International Conference Polynomial Computer … › 2018 › Book_PCA2018_.pdfISBN...

Russian Academy of Sciences

St.Petersburg Department

of Steklov Mathematical Institute

Euler International Mathematical Institute

St.Petersburg Electrotechnical University "LETI"

International Conference

Polynomial Computer Algebra

International Conference on Polynomial Computer Algebra

St. Petersburg, April, 2018

Санкт Петербург

2018

ISBN 978-5-9651-1141-1

International Conference

Polynomial Computer Algebra '2018

St. Petersburg, Russia

April 16-21, 2018

International Euler Institute

International Conference Polynomial Computer Algebra '2016; St. Petersburg,

April 18-21 2018/ Euler International Mathematical Institute, Ed. by N.N.Vassiliev,

VVM Pubishing, 2018, 121 p.

The book contains short papers, extended abstracts and abstracts of reports

presented at the International Conference on Polynomial Computer Algebra 2018,

St.Petersburg, April 2018

© St. Petersburg department of Steklov

Institute of Mathematics, RAS, 2018

Agenda

April 16

9:30 – 10:20 Registration, coffee

10:20 – 10:30 Opening the conference

10:30 – 11:10 Vladimir Gerdt, Markus Lange-Hegermann, Daniel

Robertz

Thomas decomposition of differential systems andits implementationin Maple

11:10 – 11:40 Vladimir Kornyak

Irreducible Decomposition of Representations of Finite Groupsvia Polynomial Computer Algebra

11:40 – 12:00 Coffee break

12:00 – 12:30 Ronen Peretz

A sharp version of Shimizu’s theorem on entire automorphicfunctions

12:30 – 13:00 Nikolai Proskurin

Notes on character sums and complex functions over finitefields

13:00 – 15:00 Lunch

15:00 – 15:30 Evgenii Mityushov, Natalia Misyura, Svetlana Berestova

On finite subgroups of SO(3), regular polyhedrons in R4 andthe spherical motion of rigid body

15:30 – 16:00 Alexandr Seliverstov

Real cubic hypersurfaces containing no line of singular points


16:30 – 17:00 Mikhail Babich, Sergey Slavyanov

Links between second-order Fuchsian equations and first-order linear systems

17:00 – 17:30 Aleksandr Salatich, Sergey Slavyanov

Confluent Heun Equation and equivalent first-order systems

17:30 – 18:00 Maksim Karev

Double Hurwitz Numbers

18:30 WELCOME PARTY

3

April 17

10:00 – 10:50 Lorenzo Robbiano

Computational Linear and Commutative Algebra


11:20 – 12:10 Chenqi Mou

On the chordality of polynomial sets in triangular decompositionin top-down style

12:10 – 13:00 Dima Grigoriev, Nicolai Vorobjov

Upper bounds on Betti numbers of tropical prevarieties

13:00 – 15:00 Lunch

15:00 – 15:20 Sergei Soloviev, Mark Spivakovsky, Nikolay Vassiliev

To the memory of Sergei Baranov

15:20 – 16:10 Anatoly Vershik

To make simulation on algorithm RSK for Bernoulli sequences


16:40 – 17:10 Mikhail Malykh, Leonid Sevastianov, Yu Ying

Elliptic functions and finite difference method

17:10 – 17:40 Andrei Malyutin

What does a random knot look like?

17:40 – 18:10 Vahagn Abgaryan, Arsen Khvedelidze, Astghik Torosyan

On a moduli space of theWigner quasiprobability distributions

April 18

10:00 – 10:50 Michela Ceria, Teo Mora

Combinatorics of ideals of points: a Cerlienco-Mureddu- likeapproach for an iterative lex game.

10:50 – 11:20 Alexander Chistov

A New Approach to Effective Computation of the Dimensionof an Algebraic Variety


11:50 – 12:30 Martin Kreuzer, Le Ngoc Long, Lorenzo Robbiano

On the Cayley-Bacharach Property

4

12:30 – 13:00 Alexander Batkhin

q-Analogue of discriminant set and its computation

13:00 – 15:00 Lunch

15:00 – 15:30 Dominik Michels, Vladimir Gerdt, Dmitry Lyakhov,

Yuri Blinkov

On Strongly Consistent Finite Difference Approximations

15:30 – 16:00 Aleksandr Myllari, Tatiana Myllari, Anna Myullyari,

Nikolay Vassiliev

On the complexity of trajectories in the equal-mass free-fallthree-body problem

16:00 – 16:30 Alexander Tiskin

Weighted seaweed braids


17:00 – 17:30 Darya Chemkaeva, Alexander Flegontov

Bifurcation diagrams for polynomial nonlinear ODE

17:30 – 18:00 Ioannis Parasidis, Efthimios Providas

Factorization Method for the Second-Order LinearNonlocalDifference Equations

18:30 Chamber music Concert. Euler Institute

April 19

10:00 – 10:40 Mark Spivakovsky

On the Pierce-Birkhoff conjecture and related problems.

10:40 – 11:20 David R. Stoutemyer, David J. Jeffrey, Robert M.Corless

Integration and the specialization problem


11:50 – 12:30 Nikolai Vavilov

Reverse decomposition of unipotents

12:30 – 14:00 Lunch

14:00 Bus Excursion around St.Petersburg

The bus will start from the Andersen hotel at 14:00.

19:00 BANQUET

5

April 20

10:00 – 10:40 Fedor Petrov

Arithmetic progressions in 2-groups

10:40 – 11:10 Vasilii Duzhin, Nikolay Vassiliev

Schutzenberger transformation on graded graphs: Implementationand numerical experiments.


11:40 – 12:10 Anton Chukhnov, Ilya Posov, Sergei Pozdniakov

Computer assisted constructive tasks as tasks with infiniteset of solutions for math olympiads and contests

12:10 – 12:40 Victor Edneral, Valery Romanovski

Local and Global Integrability of ODEs

12:40 – 14:30 Lunch

14:30 – 15:00 Semjon Adlaj

Back to solving the quintic, depression and Galois primes

15:00 – 15:30 Dmitry Pavlov

Usage of Automatic Differentiation in Some Practical Problemsof Celestial Mechanics

15:30 – 16:00 Dan Aksim, Dmitry Pavlov

On the Extension of Adams–Bashforth–Moulton Methodsfor Numerical Integration of Delay Differential Equationsand Application to the Moon’s Orbit

16:00 – 17:00 Free round table discussion, Closing of the Conference,

coffee

6

Table of content

Vahagn Abgaryan, Arsen Khvedelidze, Astghik Torosyan

On a moduli space of the Wigner quasiprobability distributions . . . . . . . . . . . . 10

Semjon Adlaj

Back to solving the quintic, depression and Galois primes . . . . . . . . . . . . . . . . . . 12

Dan Aksim, Dmitry Pavlov

On the Extension of Adams–Bashforth–Moulton Methods for NumericalIntegration of Delay Differential Equations and Application to the Moon’sOrbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Mikhail Babich, Sergey Slavyanov

Links between second-order Fuchsian equations and first-order linear systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Alexander Batkhin

q-Analogue of discriminant set and its computation . . . . . . . . . . . . . . . . . . . . . . . . 20

Michela Ceria, Teo Mora

Combinatorics of ideals of points: a Cerlienco-Mureddu- like approach for aniterative lex game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Darya Chemkaeva, Alexander Flegontov

Bifurcation diagrams for polynomial nonlinear ODE. . . . . . . . . . . . . . . . . . . . . . . . 29

Alexander Chistov

A New Approach to Effective Computation of the Dimension of an AlgebraicVariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Anton Chukhnov, Ilya Posov, Sergei Pozdniakov

Computer assisted constructive tasks as tasks with infinite set of solutions formath olympiads and contests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

Vasilii Duzhin, Nikolay Vassiliev

Schutzenberger transformation on graded graphs: Implementation andnumerical experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

Victor Edneral, Valery Romanovski

Local and Global Integrability of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Vladimir Gerdt, Markus Lange-Hegermann, Daniel Robertz

Thomas decomposition of differential systems andits implementation in Maple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

7

Dima Grigoriev, Nicolai Vorobjov

Upper bounds on Betti numbers of tropical prevarieties . . . . . . . . . . . . . . . . . . . . 54

Maksim Karev

Double Hurwitz Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Vladimir Kornyak

Irreducible Decomposition of Representations of Finite Groups via PolynomialComputer Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Martin Kreuzer, Le Ngoc Long, Lorenzo Robbiano

On the Cayley-Bacharach Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

Mikhail Malykh, Leonid Sevastianov, Yu Ying

Elliptic functions and finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Andrei Malyutin

What does a random knot look like? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Dominik Michels, Vladimir Gerdt, Dmitry Lyakhov, Yuri Blinkov

On Strongly Consistent Finite Difference Approximations . . . . . . . . . . . . . . . . . . 72

Evgenii Mityushov, Natalia Misyura, Svetlana Berestova

On finite subgroups of SO(3), regular polyhedrons in R4 and the sphericalmotion of rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chenqi Mou

On the chordality of polynomial sets in triangular decomposition in top-downstyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Aleksandr Myllari, Tatiana Myllari, Anna Myullyari, Nikolay

Vassiliev

On the complexity of trajectories in the equal-mass free-fall three-body problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

Ioannis Parasidis, Efthimios Providas

Factorization Method for the Second-Order LinearNonlocal DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Dmitry Pavlov

Usage of Automatic Differentiation in Some Practical Problems of CelestialMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Ronen Peretz

A sharp version of Shimizu’s theorem on entire automorphic functions . . . . . . 91

8

Fedor Petrov

Arithmetic progressions in 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Nikolai Proskurin

Notes on character sums and complex functions over finite fields . . . . . . . . . . . .97

Lorenzo Robbiano

Computational Linear and Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 104

Aleksandr Salatich, Sergey Slavyanov

Confluent Heun Equation and equivalent first-order systems . . . . . . . . . . . . . . . 105

Alexandr Seliverstov

Real cubic hypersurfaces containing no line of singular points . . . . . . . . . . . . . 109

Mark Spivakovsky

On the Pierce-Birkhoff conjecture and related problems. . . . . . . . . . . . . . . . . . . .111

David R. Stoutemyer, David J. Jeffrey, Robert M.Corless

Integration and the specialization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Alexander Tiskin

Weighted seaweed braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Nikolai Vavilov

Reverse decomposition of unipotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9

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11

Back to solving the quintic,

depression and Galois primes

Semjon Adlaj

Abstract. Évariste Galois is best known for proving the insolubility of thegeneral quintic via radicals. There, he (merely) confirmed the ingeniousinsights of Carl Gauss, Paolo Ruffini and Niels Abel. Yet, Galois went on(spectacularly alone) to formulate both necessary and sufficient criterion forsolubility of a general algebraic equation via radicals. Even more, he wasundeniably the first to actually solve the general quintic via exhibiting it asa modular equation of level 5. We aim and (hopefully) succeed at lifting anyremaining doubts, concerning the latter (persistently hardly ever known)claim. And along with presenting Galois construction for depressing thedegree of the modular equation of level 5, 7 or 11, we show that suchconstruction is unique for the (Galois) prime 5, but one more construction ispossible for each of the two remaining Galois primes 7 and 11.

In his last letter [5], eloquently described by Hermann Weyl as “the mostsubstantial piece of writing in the whole literature of mankind”, Évariste Galoisindicated sufficient and necessary condition for depressing the degree of themodular equation of prime level. For this purpose he introduced the projectivespecial linear group over a prime field, which we denote by Gp,1 and observedthat it was simple whenever the prime p strictly exceeded the prime 3.2 Hepointed out the three exceptional primes for which the group Gp possessed asubgroup of index, coinciding with p. These were the primes 5, 7 and 11. For anyprime p strictly exceeding 11 only subgroups of index p + 1, and no lower, areguaranteed to exist. Equivalently said, a modular equation, of prime level p ≥ 5,

1The group Gp might be viewed as the Galois group (in the common sense) of its correspondingalgebraic equations, as we shall further clarify. The standard notation for Gp is PSL(2,Fp), where

we assume the index p to denote a prime.2The very concept of simplicity, being introduced by Galois, is the basis for classifying groups.The classification of finite simple groups, which referred to as “an enormous theorem”, was(prematurely) announced in 1981 (by Daniel Gorenstein) before it was completed in 2004 (byMichael Aschbacher and Stephen Smith).

12

2 Semjon Adlaj

is depressable,3 from degree p + 1 to degree p, iff p ∈ 5, 7, 11. Via explicitlyconstructing the subgroups, corresponding to these three exceptional primes,Galois must, in particular, be solely credited for actually solving the generalquintic via exhibiting it as a modular equation of level 5. While Galois’contribution for formulating sufficient and necessary criterion for solubility of analgebraic equation via radicals is acknowledged, his decisive contribution toactually solving the quintic (before Hermite and Klein) is, surprisingly, toopoorly recognized (if not at all unrecognized)! Betti, in 1851 [3], futily askedLiouville not to deprive the public any longer of Galois’ (unpublished) results,and, in 1854 [4], went on to show that Galois’ construction yields a solution tothe quintic via elliptic functions.4 One might associate with each quintic, given inBring-Jerrard form, a corresponding value for the (Jacobi) elliptic modulus β, asHermite did, in 1958 [6], implementing this very Galois’ construction (therebyenabling an efficient algorithm for calculating the roots via values of an ellipticfunction at points placed apart by multiples of fifth-period). The group G5 acts(naturally) on the projective line PZ5, which six elements we shall, followingGalois, label as 0,1,2,3,4 and ∞. Then collecting them in a triple-pair(0, ∞),(1, 4),(2, 3), the group G5 is seen to generate four more triple-pairs(1, ∞),(2, 0),(3, 4),(2, ∞),(3, 1),(4, 0),(3, ∞),(4, 2),(0, 1),(4, ∞),(0, 3),(1, 2).Together, the five triple-pairs constitute the five-element set upon which G5

acts.5 Galois did not (in his last letter) write down the four triple-pairs, which wedid write after the first, and we now, guided by his conciseness and brevity,confine ourselves to writing down only the first pair-set that he presented foreach of the two remaining cases, where p = 7 and p = 11, respectively:(0, ∞),(1, 3),(2, 6),(4, 5) and (0, ∞),(1, 2),(3, 6),(4, 8),(5, 10),(9, 7). Unlike thecase p = 5, an alternative might be presented for the case p = 7, which is(0, ∞),(1, 5),(2, 3),(4, 6), and for the case p = 11, which is (0, ∞),(1, 6),(3, 7),(4, 2),(5, 8),(9, 10). The “absolute invariant” for the action of the subgroup Γ2,of the modular group Γ := PSL(2,Z), consisting of linear fractionaltransformations congruent to the identity modulo 2, is the square (of the elliptic

3This well-established term means lowerable. Its conception is a simple (yet ingenious) idea withwhich Galois alone must be fully credited, and, as we shall soon see, is the single most crucial

(yet rarely brought to awareness) step towards actually solving the quintic.4In 1830, Galois competed with Abel and Jacobi for the grand prize of the French Academy ofSciences. Abel (posthumously) and Jacobi were awarded (jointly) the prize, whereas all referencesto Galois’ work (along with the work itself!) have (mysteriously) disappeared. The very fact thatGalois’ lost works contained contributions to Abelian integrals is either unknown (to many) ordeemed (by some) no longer relevant to our contemporary knowledge. For the sake of being fair

to a few exceptional mathematicians, we must cite (without translating to English) Grothendick

(as a representative), who (in his autobiographical book Récoltes et Semailles) graciously admitsthat “Je suis persuadé d’ailleurs qu’un Galois serait allé bien plus loin encore que je n’ai été.D’une part à cause de ses dons tout à fait exceptionnels (que je n’ai pas reçus en partage, quantà moi).”5Indeed, it is the five-element set (not merely a five-element set) which Hermite had no choicebut to employ. Galois’ construction for each of the two remaining cases, where p = 7 or p = 11,

allows an alternative, as will, next, be exhibited.

13

Back to solving the quintic, depression and Galois primes 3

modulus) β2. A fundamental domain Γ2\H, for the action of Γ2 (on the upperhalf-plane H), might be obtained by subjecting a fundamental domain Γ\H (ofΓ) to the action of the quotient group Γ/Γ2

∼= S3.6 In particular, β2 viewed asfunction on H, is periodic, with period 2. Sohnke, in a remarkable work [7], haddetermined the modular equations for β1/4, for all odd primes up to, andincluding, the prime 19. That work, along with Betti’s work, inspired Hermite to(successfully) relate a (general) quintic, in Bring-Jerrard form, to a modularequation of level 5, yet he had little choice but to admit the importance of a soleGalois idea (in depressing the degree of the modular equation).7 The modularpolynomial for β1/4, of level 5, is

φ5(x, y) := x6 − y6 + 5 x2y2 (x2 − y2) + 4 x y (1 − x4y4),

and the period of β1/4 (as an analytically continued function) is 16. Denoting theroots of φ5(x, y = β1/4(τ)), for a fixed τ ∈ H, by

y5 = β1/4(5 τ), ym = −β1/4

(

τ + 16 m

5

)

, 0 ≤ m ≤ 4,

one calculates the minimal polynomial for x1 := (y5 − y0)(y4 − y1)(y3 − y2) y. Itturns out to be

x5 − 2000 β2 (1 − β2)2 x + 1600√

5 β2 (1 − β2)2 (1 + β2).

Thereby, a root of the quintic

x5 − x + c, c :=2 (1 + β2)

55/4√

β(1 − β2)=

2 (1 + y8)

55/4 y2√

1 − y8, 8

is √5 c x1

4 (1 + β2)=

x1

2√

5√

5 β(1 − β2)=

(y5 − y0)(y4 − y1)(y3 − y2)

2 y√

5√

5 (1 − y8),

6The latter quotient group coincides with G2 which is isomorphic with S3.7Hermite had apparently adopted Cauchy’s catholic and monarchist ideology, much in contrastto Galois’ passionate rejection of social prejudice. In 1849, Hermite submitted a memoir to

the French Academy of Sciences on doubly periodic functions, crediting Cauchy, but a prioritydispute with Liouville prevented its publication. Hermite was then elected to the French Academy

of Sciences on July 14, 1856, and (likely) acquainted, by Cauchy, with ideas stemming from

(but not attributed to) Galois “lost” papers. T. Rothman made a pitiful attempt in “Geniusand Biographers: The Fictionalization of Evariste Galois”, which appeared in the AmericanMathematical Monthly, vol. 89, 1982, pp. 84-106 (and, sorrowly, received the Lester R. FordWriting Award in 1983) to salvage Cauchy’s reputation (unknowingly) suggesting further evidenceof Cauchy’s cowardice, and surprising us, along the way, with many (unusual but ill substantiatedand biased) judgements telling us much about T. Rothman himself, but hardly anything

trustworthy about anyone else!8One must note that the constant coefficient c is invariant under the inversions β Ô→ −1/β and

β Ô→ (1 − β)/(1 + β). Here, the composition of the latter two inversions is another inversion.The corresponding four-point orbit in a fundamental domain Γ2\H is generated via the mappingτ Ô→ 2/(2 − τ).

14

4 Semjon Adlaj

and so is expressible via the coefficients λm and µm of the elliptic polynomialsrm5(x) = x2 − λmx + µm, 0 ≤ m ≤ 5.9 In fact, the polynomials rm5 might beso ordered so that, for each m, the value β2

m coincides with y8

m. The (general)expression for y8

m = β2

m might be written as

y8

m =s(λm, µm, β)

β4s(λm, µm, 1/β),

where

s(λ, µ, x) =

(

1 + λ x

µ+ x2

) (

4 λ +

(

2 λ2

µ+ 4 + 5 µ

)

x + λ

(

2

µ+ 3

)

x2 + x3

)

,

and the coefficients λm = γm + (2 · γm) and µm = γm(2 · γm) satisfy5

∏

m=0

(

x2 − λm x + µm

)

= x12 +62 x10

5− 21 x8 − 60 x6 − 25 x4 − 10 x2 +

1

5+

+ α x3

(

x8 + 4 x6 − 18 x4 − 92 x2

5− 7

)

+ α2x4

(

x6

5− 3 x2 − 2

)

− α3x5

5= r5(x),

where α := 4(β + 1/β). The roots γm and 2 · γm,10 0 ≤ m ≤ 5, of the divisionpolynomial r5 might be highly efficiently calculated via the algorithm provided in[1]. Calculating a pair, say γ0 and γ5, suffices, of course, for calculating all twelveroots via applying the addition formula along with the doubling formula, as toldin [2].

Nowadays, oblivion has entirely replaced marvelling at Galois key step,towards solving the quintic, in depressing the degree of the modular equation, oflevel 5, from 6 to 5,11 and Galois is merely mentioned, along with Abel, fordetermining that the quintic is not solvable via radicals. We hope that this(crippled) view of Galois (deeply constructive) theory would finally come to anend.

References

[1] Adlaj S. Iterative algorithm for computing an elliptic integral // Issues on motionstability and stabilization (2011), 104-110 (in Russian).

[2] Adlaj S. Multiplication and division on elliptic curves, torsion points and roots

of modular equations. Available at http://www.ccas.ru/depart/mechanics/TUMUS/

Adlaj/ECCD.pdf.

9The elliptic polynomials were presented, in 2014, at the 7th PCA conference(http://pca.pdmi.ras.ru/2014/program) in a talk titled “Modular polynomial symmetries”, and

at the 17th workshop on Computer Algebra (http://compalg.jinr.ru/Dubna2014/abstracts.html)in a talk titled “Elliptic and coelliptic polynomials”. Details are provided in [2].10Consistently with the notation employed in [2], 2 · γm signifies that the doubling formula hasbeen applied to γm.11For example, S. VlăduŃ (wrongfully) attributes, in his book “Kronecker’s Jugendtraum andModular Functions” (published by Gordon and Breach in 1991), to Hermite showing theequivalence of the general quintic to the modular equation of level 5.

15

Back to solving the quintic, depression and Galois primes 5

[3] Betti E. Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado

primo // Dagli Annali di Scienze matimatiche e fisiche, t. II (Roma, 1851): 5-19.[4] Betti E. Un teorema sulla risoluzione analitica delle equazioni algebriche // Dagli

Annali di Scienze matimatiche e fisiche, t. V (Roma, 1854): 10-17.

[5] Galois É. “Lettre de Galois à M. Auguste Chevalier” // Journal de MathématiquesPures et Appliquées XI (1846): 408–415.

[6] Hermite C. “Sur la résolution de l’équation du cinquième degré” // Comptes Rendusde l’Académie des Sciences XLVI(I) (1858): 508–515.

[7] Sohnke L. Equationes Modulares pro transformatione functionum Ellipticarum //Journal de M. Crelle, t. XVI (1836): 97-130.

Semjon AdlajSection of Stability Theory and Mechanics of Controlled SystemsDepartment of MechanicsDivision of Complex Physical and Technical Systems ModelingComputing Center of the Federal Research Center “Informatics and Control”Russian Academy of SciencesRussia 119333, Moscow, Vavilov Street 40.e-mail: [email protected]

16

On the Extension of Adams–Bashforth–Moulton

Methods for Numerical Integration of Delay Dif-

ferential Equations and Application to the Moon’s

Orbit

Dan Aksim and Dmitry Pavlov

Abstract. One of the problems arising in modern celestial mechanics is theneed of precise numerical calculation of the Moon’s orbit. Due to the nature oftidal forces, their action is modeled with a time delay and the orbit is thereforedescribed by a so-called delay differential equation (DDE). Numerical inte-gration of the orbit is normally being performed in both directions (forwardsand backwards in time) from some epoch, and while the theory of normalforward-in-time numerical integration of DDEs is developed and well-known,integrating a DDE backwards in time is equivalent to solving a special kindof DDE called an advanced-delay differential equation, where the derivativeof the function depends on not yet known future states of the function, whichpresents a certain numerical challenge.

The present work examines a modification of Adams–Bashforth–Moultonmethod allowing to perform integration of the Moon’s DDE forwards andbackwards in time and the results of such integration.

Dan AksimLaboratory of Ephemeris AstronomyInstitute of Applied Astronomy of the Russian Academy of SciencesSt. Petersburg, Russiae-mail: [email protected]

Dmitry PavlovLaboratory of Ephemeris AstronomyInstitute of Applied Astronomy of the Russian Academy of SciencesSt. Petersburg, Russiae-mail: [email protected]

17

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19

q-Analogue of discriminant set and its computa-

tion

Alexander Batkhin

Abstract. A generalization of the classical discriminant of the polynomialwith arbitrary coefficients defined using the linear Hahn operator that de-creases the degree of the polynomial by one is studied. The structure of thegeneralized discriminant set of the real polynomial, i.e., the set of values ofthe polynomial coefficients at which the polynomial and its Hahn operatorimage have a common root, is investigated. The structure of the generalizeddiscriminant of the polynomial of degree n is described in terms of the parti-tions of n. Algorithms for the construction of a polynomial parameterizationof the generalized discriminant set in the space of the polynomial coefficientsare proposed. The main steps of these algorithms are implemented in a Maplelibrary.

Introduction

Let g : R → R : x 7→ g(x) be a given smooth one-to-one map of the real axis,which is the domain of polynomial f(x) with arbitrary coefficients. We want tofind conditions on the coefficients of the polynomial under which it has at least apair of roots ti, tj satisfying the relation g(ti) = tj and investigate the structure ofthe algebraic variety in the space of coefficients possessing such property.

Here we consider a generalization of the classical discriminant of the poly-nomial. This generalization naturally includes the classical discriminant and itsanalogs emerging when the q-differential and difference operators that have a well-developed calculus [1] and important applications [2] are used. It turned out thatthe constructs that were earlier obtained for investigating the discriminant [3] andresonance sets [4] can be extended for a more general case.

The aim of this research is to propose an efficient algorithm for calculatingthe parametric representation of all components of the g-discriminant set Dg(f)of the monic polynomial f(x).

20

2 Alexander Batkhin

1. Generalized discriminant set

Definition 1. Define the q-bracket [a]q, q-Pochhammer symbol (a; q)n, q-factorial[n]q!, q-binomial coefficients (Gaussian) coefficients

[

n

k

]

qas follows:

[a]q =qa − 1

q − 1, a ∈ R\0, (a; q)n =

n−1∏

k=0

(

1− aqk)

, (a; q)0 = 1,

[n]q! =

n∏

k=1

[k]q =(q; q)n(1− q)n

, q 6= 1,

[

n

k

]

q

=[n]q!

[n− k]q! [k]q!=

k∏

i=1

qn−i+1 − 1

qi − 1.

As q → 1, all these objects become classical.Define a g-analogue of the standard binomial (x− a)n so called g-binomial

x; tn;g ≡

n−1∏

i=0

(x− gi(t)), x; t0;q = 1.

Here gk is the k-th iteration of the diffeomorphism g, k ∈ Z (see below).

Let fn(x) be is a monic polynomial of degree n with complex coefficientsdefined by

fn(x)def= xn + a1x

n−1 + a2xn−2 + · · ·+ an.

Let P be the space of polynomials over R and let g

g : R→ R : x 7→ qx+ ω, q, ω ∈ R, q 6= −1, 0,

be a linear diffeomorphism on R that induces a linear Hahn operator Ag on P,satisfying the following two conditions:

1. the degree reduction: deg(Ag fn)(x) = n− 1; in particular, Ag x = 1;2. Leibnitz rule analogue:

(Ag xfn)(x) = fn(x) + g(x)(Ag fn)(x).

The Hahn operator Ag called below g-derivative has the form

(Ag f)(x)def=

f(qx+ ω)− f(x)

(q − 1)x+ ω, x 6= ω0,

f ′ (ω0) , x = ω0,(1)

where ω0 = ω/(1− q) is the fixed point of g. Parameters q and ω are satisfied thefollowing conditions q, ω ∈ R, q 6= −1, 0 and (q, ω) 6= (1, 0). The g-derivative Ag

can be considered as a generalization of the q-differential Jackson operator Aq atω = 0, q 6= 1, as the difference operator ∆ω at q = 1 and as the classical derivatived/dx in the limit q → 1 and ω = 0.

q-Analogs of many mathematical objects emerged already in Euler’s works,and then were elaborated by many mathematicians (see the historical review in [1]).The q-calculus has recently became a part of the more general construct calledquantum calculus [5]. It has numerous applications in various fields of modernmathematics and theoretical physics. For example, many applications related to

21

q-Analogue of discriminant set 3

the theory of orthogonal polynomials and their various generalizations, it is im-portant to determine the conditions on the coefficients ai, i = 1, . . . , n, of thepolynomial fn(x) under which it has roots satisfying g(ti) = tj .

Definition 2. The pair of roots ti, tj , i, j = 1, . . . , n, i 6= j of the polynomial fn(x)is said to be g-coupled if g(ti) = tj .

Let consider the following problem.

Problem. In the coefficient space Π ≡ Cn of the polynomial fn(x), investigate the

g-discriminant set denoted Dg(fn) on which this polynomial has at least one pairof g-coupled roots.

Definition 3. The sequence Seq(k)g (t1) of g-coupled roots of length k is defined asthe finite sequence ti, i = 1, . . . , k in which each term, beginning with the secondone, is a g-coupled root of the preceding term: g(ti) = ti+1. The initial root t1 is

called the generating root of the sequence Seq(k)g (t1) .

For each fixed set of parameters q, ω, the g-discriminant set Dg(fn) consists

of a finite set of varieties Vk on each of which fn(x) has k sequences Seq(li)g (ti)

of g-coupled roots of length i with different generating roots ti, i = 1, . . . , k. Toobtain an expression for the generalized (sub)discriminant of the polynomial fn(x)in terms of its coefficients, any method available in the classical elimination theorycan be used. If we replace the derivative f ′

n(x) by the polynomial Ag fn(x), thenany matrix method for calculating the resultant of a pair of polynomials gives an

expression of the generalized k-th subdiscriminant D(k)g (fn) (see [6, 3] for details).

Theorem 1. The polynomial fn(x) has exactly n−d different sequences of g-coupled

roots, iff the first nonzero element in the sequence of i-th generalized subdiscrimi-

nants D(i)g (fn) is the subdiscriminat D

(d)g (fn) with the index d.

2. Algorithm of parametrization of Dg(fn) and its implementation

Definition 4. The partition λ of a natural number n is any finite nondecreasing

sequence of natural numbers λ1 ≤ λ2 ≤ · · · ≤ λk, for which∑k

i=1 λi = n. Eachpartition λ will be written as λ = [1n12n23n3 . . . ].

Consider the partition λ = [1n12n23n3 . . . ] of the natural number n. Thequantity i in the partition λ determines the length of the sequence of g-coupledroots for the corresponding generating root ti, and ni is the number of differentgenerating roots determining the sequence of roots of length i. Every partitionλ of n determines the structure of the g-coupled roots of the polynomial fn(x),and this structure is associated with the algebraic variety Vi

l , i = 1, . . . , pl(n) ofdimension l corresponding to the number of different generating roots ti in thecoefficient space Π.

22

4 Alexander Batkhin

Consider the partition[

n1]

corresponding to the case when there is a uniquesequence of roots of length n specified by the generating root t1. Then, the poly-nomial fn(x) is a g-binomial x; t1n;g and its coefficients ai can be representedin terms of the elementary symmetric polynomials σi(x1, x2, . . . , xn) calculated onthe roots gj(t1), j = 0, . . . , n− 1,

ai = (−1)iσi

(

t1, g(t1), . . . , gn−1(t1)

)

, i = 1, . . . , n.

Let consider the polynomial fn(x) ≡ x; t1n;g with the structure of rootscorresponding to the partition

[

n1]

. Using [7, Lemmas 2, 3], we conclude that, forevery k such that 0 < k ≤ n, it holds that

k∑

i=0

[

n

i

]

q

[n− i]q!

[n]q!

(

Agi fn

)

(t1)t2; t1i;g = fk(x; t2) · fn−k(x; gk(t1)), (2)

where (Ag0 f)(x) ≡ f(x). Therefore, formula (2) allows us to pass from the polyno-

mial with the structure of roots corresponding to the partition[

n1]

to a polynomial

with the structure of roots determined by the partitions[

k1(n− k)1]

or[

(n/2)2]

,if k = n/2.

Theorem 2. Let there be a variety Vl, dimVl = l on which the polynomial fn(x)

has different sequences of g-coupled roots and the sequence of roots Seq(m)g (t1) has

length m > 1. The roots of the other sequences are not g-coupled with all roots of

the sequence Seq(m)g (t1). Let rl(t1, . . . , tl) be a parameterization of the variety Vl.

Then for 0 < k < n, the formula

rl(t1, . . . , tl, tl+1) = rl(t1, . . . , tl) +k

∑

i=1

[

k

i

]

q

[m− i]q!

[m]q!

(

Agirl

)

(t1)tl+1; t1i;g

specify a parameterization of the part of Vl+1 on which there are two sequences of

roots Seq(m−k)g (gk(t1)) and Seq(k)g (g(tl+1)), and the other sequences of roots are

the same as on the original variety Vl.

We introduce two basic operations that allow us to successively pass fromthe parametric representation of the one-dimensional variety V1 to the parameter-ization of all other components of the g-discriminant set Dg(fn).

1. The operation of passing from the variety Vl to the variety Vl+1 in Theorem 2is called ASCENT of order k. If fn(x) has only real roots on this variety, thenwe obtain its complete parameterization; if there are complex roots, then weapply the following operation.

2. The operation called CONTINUATION makes it possible to obtain a param-eterization of the entire variety Vl+1 obtained by the ASCENT operation inthe case when there are complex conjugate roots on it.

At each step of this algorithm, we remain within polynomial parameteriza-tions; therefore, the following result holds.

23

q-Analogue of discriminant set 5

Proposition. For fixed values of parameters (q, ω) of the Hahn operator (1), theg-discriminant set Dg(fn) of the polynomial fn(x) admits a polynomial parame-terization of each of the algebraic varieties Vk

l , l = 1, . . . , n − 1, k = 1, . . . , pl(n),that form this set.

For calculating the g-discriminant set Dg(fn), a number of procedures inMaple and Sympy were developed. Their description and application to some ex-amples are given in [8].

References

[1] T. Ernst. A Comprehensive Treatment of q-Calculus. Springer, Basel Heidelberg NewYork Dordrecht London, 2012. http://dx.doi.org/10.1007/978-3-0348-0431-8

doi:10.1007/978-3-0348-0431-8.

[2] R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric Orthogonal Poly-

nomials and Their q-Analogues. Springer-Verlag, Berlin Heidelberg, 2010. http:

//dx.doi.org/10.1007/978-3-642-05014-5 doi:10.1007/978-3-642-05014-5.

[3] A. B. Batkhin. Parameterization of the discriminant set of a polynomial. Pro-

gramming and Computer Software, 42(2):65–76, 2016. http://dx.doi.org/10.1134/S0361768816020031 doi:10.1134/S0361768816020031.

[4] A. B. Batkhin. Structure of the resonance set of a real polynomial. Keldysh Insti-

tute Preprints, (29), 2016. (in Russian). URL: http://www.keldysh.ru/papers/2016/prep2016_29.pdf, http://dx.doi.org/10.20948/prepr-2016-29 doi:10.20948/

prepr-2016-29.

[5] V. Kac and P. Cheung. Quantum Calculus. Springer-Verlag, New York, Heidelber,Berlin, 2002.

[6] Gathen, J. von zur and T. Lücking. Subresultants revisited. Theoretical Computer

Science, 297:199–239, 2003. http://dx.doi.org/10.1016/S0304-3975(02)00639-4doi:10.1016/S0304-3975(02)00639-4.

[7] A. B. Batkhin. Parameterization of a set determined by the generalized discriminantof a polynomial. Programming and Computer Software, 44(2):75–85, 2018. http://dx.doi.org/10.1134/S0361768818020032 doi:10.1134/S0361768818020032.

[8] A. B. Batkhin. Computation of generalized discriminant of a real polynomial.Keldysh Institute Preprints, (88), 2017. (in Russian). URL: http://www.keldysh.ru/papers/2017/prep2017_88.pdf, http://dx.doi.org/10.20948/prepr-2017-88

doi:10.20948/prepr-2017-88.

Alexander BatkhinDepartment of Singular ProblemsKeldysh Institute of Applied Mathematics of RASDepartment of Theoretical MechanicsMoscow Institute of Physics and TechnologyMoscow, Russiae-mail: [email protected]

24

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25

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27

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28

Bifurcation diagrams for polynomial nonlinear

ordinary differential equations

Daria Chemkaeva and Alexandr Flegontov

Abstract. This study considers the general case for classes of nonlinear bound-ary value problems for a second-order autonomous ordinary differential equa-tion with homogeneous boundary conditions. The general case is studied ap-plying to polynomial-like nonlinearities. We investigate the number of positivesolutions to the problem. The research is confirmed by computer-generatedfunction of P. Korman, Y. Li, T. Ouyang Theorem and bifurcation diagrams.

Introduction

We study the existence of positive solutions of the nonlinear two-point boundaryvalue problem:

y′′xx + λf(y(x)) = 0, x ∈ (−1; 1), (1)

y(−1) = y(1) = 0. (2)

Assume f = f(y) so second order ODE is autonomous, where parameter λ

is positive. In this case, the bifurcation arises when the number of solutions of thedifferential equation changes as the parameter λ changes.

The problem (1)–(2) describes many physical processes, for example, belongsto the problems of combustion of gases and population dynamics. The nonlinearityof f = f(y) in combustion theory denotes intermediate steady states of the tem-perature distribution y, and the bifurcation parameter λ determines the amountof unburnt substance.

Section 1 is technical and contains useful supplement of P. Korman, Y. Li andT. Ouyang theorem. Section 2 is the main part of the study where the behaviorof function from P. Korman, Y. Li and T. Ouyang Theorem is studied consideringthat nonlinear function is a polynomial of odd degree with a2n−1 roots (n =2, . . . , k, k ≥ 2). The examples are provided with description of the respective time-map functions, solutions, bifurcations and visualizations. Finally, we summarizethe results and make conclusions.

29

2 Daria Chemkaeva and Alexandr Flegontov

1. P. Korman, Y. Li and T. Ouyang Theorem

The differential equation (1) with boundary conditions (2) has k zeros of solutionsdepending on the bifurcation parameter. Let us consider the case when the numberof zeros of the solutions is even. In this case, solutions (1)–(2) are symmetric withrespect to x = 0 [1], hence (1)–(2) can be reduced to the form:

y′′xx + λf(y(x)) = 0, x ∈ (0; 1), (3)

y′x(0) = 0, y(1) = 0. (4)

It is known that positive solutions can be determined using term y(0) = a.This zero function is a time-map for solutions (1)–(2) [2] and the maximal value ofthe solution of the boundary value problem, which uniquely determines the pair(λ, y(x)). We show that by defining a we can uniquely determine the appropriatevalue λ > 0 and the solution of the problem y = y(x).

Suppose that t =√λx so for the function y = y(t) we consider the interme-

diate Cauchy problem:

y′′tt + f(y) = 0, (5)

y′t(0) = 0, y(0) = a. (6)

We use the substitution and find the first integral of equation (5), fulfilling thefirst boundary condition (6):

y′t =√2√

F (a)− F (y), F (y) =

y∫

0

f(y) dy.

For the existence of the solution (5)–(6) it is necessary to satisfy the inequalityF (a) ≥ F (y). The solution of boundary value problem (5)–(6) in implicit form is:

t =

y∫

0

dt√

F (a)− F (t).

Returning to boundary value problem (3)–(4), the bifurcation parameter is:

λ(a) =1

2

[

a∫

0

dt√

F (a)− F (t)

]2

. (7)

The function λ = λ(a) is called the bifurcation curve; its turning points arebifurcation points. The plot of this function is called the bifurcation diagram [3],implying an image of the change in the possible dynamic modes of the system witha change in the value of bifurcation parameter λ.

The authors P. Korman, Y. Li and T. Ouyang prove that a solution of theproblem (1)–(2) with the maximal value a = y(0) is singular if and only if

G(a) ≡√

F (a)

a∫

0

f(a)− f(τ)

[F (a)− F (τ)]3/2dτ − 2 = 0, (8)

30

Bifurcation diagrams for polynomial nonlinear ordinary differential equations 3

where F (y) =

y∫

0

f(t) dt.

2. Nonlinearity as a polynomial of odd degree

Now we study the general case, assuming that the function f(y) is a polynomial,and consequently can change the sign.

We set

f(y) = (y − a1)(y − a2)(y − a3)...(y − a2n−2)(a2n−1 − y), (9)

where 0 < a1 < a2 < ... < a2n−2 < a2n−1 – isolated zeros of function f(y), i. e.f(ai) = 0. It is obviously that problem (1)–(2) has trivial solutions:

y = ai, i = 1, 2, . . . , 2n− 1. (10)

Here f(y) is a polynomial of odd degree, so it has odd number of zeros.The function (9) is negative on (a1, a2), then the function is positive on (a2, a3) .Therefore, the function has n pairs of humps, where f(y) > 0 on (a2n−2, a2n−1)and f(y) < 0 on (a2n−3, a2n−2).

We suppose that f(y) satisfies the conditions F (a1) < F (a2) . . . < F (a2n−2) <F (a2n−1). Each solution branch has its maximal values inside a single positive

hump, and, f. e., that it is necessary to have

a3∫

a1

f(y) dy > 0 in order for solutions

with maximal values in (a2, a3) to exist. Plots of functions f(y) and F (y) aredepicted on Fig. 1.

f(y)

F(y)

a1 a2 a3 a4 a5

Figure 1. f(y) and F (y)

Figure 2 shows a plot of the function (8) in the plane (a;G(a)), where f

corresponds to (9). It follows from the plot that G(a) has zeros only in the inter-vals (a2, a3), (a4, a5), (a6, a7), . . . , (a2n−2, a2n−1) therefore only in these intervals

31


bifurcation points exist. Also it is clearly that function G(a) exists only on theintervals where f(y) > 0.

a1 a2 1 a3 a4 2 a5 a6 3 a7

a

G

Figure 2. G(a) for f(y) - polynomial of odd degree

We will use the asymptotic behavior of G(a) to make the intervals, whereG(a) = 0, more precise:

1. lima→a−

2n−1

G(a) = −∞, i. e., to the left of a2n−1 there is no bifurcation point,

where n = 2, . . . , k, k ≥ 2.

2. Let exist a point σn−1 ∈ (a2n−2, a2n−1), such that

σn−1∫

a2n−3

f(s) ds = 0, so

lima→σ+

n−1

G(a) = +∞, i. e., to the right of σn−1 there is no bifurcation point, where

n = 2, . . . , k, k ≥ 2.

Polynomial of third degree (cubic) is well-studied in [4]. Authors show theexistence of a critical value of the parameter λ = λ0, so that for 0 < λ < λ0 theproblem (3)–(4) with f(y) = (y− a1)(y− a2)(a3 − y) has exactly one solution, forλ = λ0 it has exactly two solutions, and exactly three solutions for λ > λ0.

Let consider the examples of polynomials of fifth and seventh degrees.

Example 1. Polynomial of 5th degree.

Let f(y) = (y−1)(y−2)(y−4)(y−5)(7−y) [5]. First, we plot f(y) and F (y) (3(a))and G(a) (3(b)) to visualize their behavior. It follows from the plot of the functionG(a) that there are two bifurcation points on intervals (a2, a3), and (a4, a5), wheref(y) > 0 (intervals (2; 4) and (5; 7)). We plot a bifurcation diagram as it presentedin formula (7) corresponding to this problem (Fig. 4).

32


F(y)

f(y)1 2 3 4 5 6 7

0

50

100

150

200

(a) f(y) and F (y)

1 2 3 4 5 6 7

-100

-50

50

100

(b) G(a)

Figure 3. Plots of f(y), F (y) and G(a) for example 1

Using accurate commands NMinimize and FindRoot of Wolfram Mathemat-ica we define turning points of λ(a): a1 ≈ 3.2417, a2 ≈ 6.5866, where λ0 ≈ 0.56973and λ1 ≈ 0.6321 (they are sorted in ascending order).

1

0

2 4 6 8

a0.0

0.5

1.0

1.5

Figure 4. λ(a) for problem in example 1

As we see on Fig. 4 there exists 0 < λ0 < λ1 such that for λ < λ0 thereis one solution, λ = λ0 there are two solutions, for λ0 < λ < λ1 there are threesolutions, for λ = λ1 there are four solutions and λ > λ1 there are five solutionsto the problem (1)–(2), where f(y) = (y − 1)(y − 2)(y − 4)(y − 5)(7− y).

Example 2. Polynomial of 7th degree.

Let f(y) = (y − 1)(y − 2)(y − 4)(y − 5)(y − 7)(y − 8)(10− y). Again we plot f(y)and F (y) (Fig. 5(a)) and G(a) (Fig. 5(b)) to define the intervals where bifurcationpoints can occur. There are three bifurcation points, each on interval (a2, a3),(a4, a5) and (a6, a7), respectively. These intervals are (2; 4), (5; 7) and (8; 10),where f(y) > 0. Bifurcation diagram for this problem is presented at Fig. 6.With the help of numeric computing methods of Wolfram Mathematica we define

33


f(y)

F(y)

2 4 6 8 10

0

2000

4000

6000

8000

10000

12000

(a) f(y) and F (y)

1 2 3 4 5 6 7 8 9 10

a

G

(b) G(a)

Figure 5. Plots of f(y), F (y) and G(a) for example 2

bifurcation points: a1 ≈ 3.27276, a2 ≈ 6.38791, a3 ≈= 9.6693, where ordered byascending values of λ are λ0 ≈ 0, 0181, λ1 ≈ 0.0194, λ2 ≈ 0.0633.

Figure 6. λ(a) for problem in example 2

There exist 0 < λ0 < λ1 < λ2 such that for λ < λ0 there is one solutionto the problem, λ = λ0 there are two solutions, for λ0 < λ < λ1 there are threesolutions, for λ = λ1 there are four solutions, λ1 < λ < λ2 there are five solutions,λ = λ2 there are six solutions and λ > λ2 there are seven solutions to the problem.

Conclusion

The obtained results generalize [5]. The study of the function G(a) showed thatit has zeros only in the intervals (a2, a3), (a4, a5), (a6, a7), . . . , (a2n−2, a2n−1),where f(y) – polynomial of odd degree (f(ai) = 0), and consequently only theseintervals contain bifurcation points. The odd degree of the polynomial f(y) exactlydetermine the number of solutions of BVP (1)–(2). The bifurcation approach to

34


the problem assists to find out bifurcation parameters λi to understand when thenumber of solutions changes.

Computational methods of numerical integration and differentiation, as wellas visualization of G(a) and λ(a) in the computing system Wofram Mathematica11.0, have defined themselves as an effective tool for studying the function G(a)from P. Korman, Y. Li and T. Ouyang Theorem, bifurcation curves and findingout the number of positive solutions of the problem.

References

[1] P. Korman, Global Solution Branches and Exact Multiplicity of Solutions for Two

Point Boundary Value Problems. Handbook of Differential Equations: Ordinary Dif-ferential Equations, 2006.

[2] R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, LectureNotes in Mathematics, Springer-Verlag, 1990.

[3] P. Korman, Y. Li, T. Ouyang Computing the location and the direction of bifurcation,Math. Research Letters, 2005.

[4] P. Korman, Y. Li, T. Ouyang Exact multiplicity results for boundary value problems

with nonlinearities generalising cubic, Proc. Royal Soc. Edinburgh, 1996.

[5] P. Korman, Y. Li, T. Ouyang Verification of bifurcation diagrams for polynomial-like

equations, Journal of Computational and Applied Mathematics, 2008.

Daria ChemkaevaDep. of Informatics & TechnologyHerzen State Pedagogical UniversitySt.Petersburg, Russiae-mail: [email protected]

Alexandr FlegontovDep. of Informatics & TechnologyHerzen State Pedagogical UniversitySt.Petersburg, Russiae-mail: [email protected]

35

A New Approach to Effective Computation of the

Dimension of an Algebraic Variety

Alexander L. Chistov

Abstract. We discuss a new method for computing the dimension of an alge-braic variety. It is based on the effective version of the first Bertini theoremfor hypersurfaces suggested by the author earlier.

Computation of the dimension of an algebraic variety is a classical problemin effective algebraic geometry. In the most simple case it is formulated as follows.Let k be a field with the algebraic closure k. Given homogeneous polynomialsf1, . . . , fm ∈ k[X0, . . . , Xn] the problem is to compute the dimension of the alge-braic variety Z(f1, . . . , fm) of all the common zeroes of the polynomials f1, . . . , fmin the projective space P

n(k).Assume additionally that the degrees degX0,...,Xn

fj 6 d for an integer d > 2for all 1 6 i 6 m. Then the number of coefficients of each polynomial fj is at most(

n+d

n

)

. So it is bounded from above by a polynomial in dn.On the other hand, one can verify whether the set Z(f1, . . . , fm) is finite (or

empty) and if #Z(f1, . . . , fm) < +∞ solve the homogeneous system f1 = . . . =fm = 0 over the algebraically closed field k. The complexity of this algorithm ispolynomial in dn and the size of the input data, see [4]. Actually the main ideasfor solving homogeneous systems of polynomial equations with a finite number ofroots are classical and were known at the beginning of the previous century, see[5].

Let us return to the general case. Now the probabilistic algorithm for com-puting the dimension of an algebraic variety is simple. Let s be an integer suchthat −1 6 s 6 n. Let us choose linear forms L0, . . . , Ls ∈ k[X0, . . . , Xn] randomly.Then the dimension dimZ(f1, . . . , fm) is the least s such that the set

Z(f1, . . . , fm, L0, L1, . . . , Ls)

is empty. So one can compute the dimension of a projective algebraic varietyprobabilistically within the time polynomial in dn and the size of the input data.

But to compute the dimension deterministically is much more difficult. Inthe case of arbitrary characteristic it is an open problem

36

2 Alexander L. Chistov

(*) to construct a deterministic algorithm for computing the dimension of a pro-jective algebraic variety Z(f1, . . . , fm) with bitwise complexity polynomial indn and the size of the input data.

We think that for arbitrary characteristic of the ground field this problem will notbe solved in near future (say, in this century).

Still here there have been a major progress. In the case of the ground field ofzero–characteristic we solved the problem (*), see [1]. We could obtain the mainresult of [1] using the methods of real algebraic geometry. After that we havedeveloped the whole theory basing on these methods and get many importantresults. However, to many specialists it seemed unnatural to apply the methods ofreal algebraic geometry for varieties over algebraically closed fields. On the otherhand, it is a fact that all other attempts to compute the dimension deterministicallywithin the time polynomial in dn and the size of the input data have been fruitless.

The situation has changed after the results of [2]. Namely, in [2] we got avery strong and explicit version of the first Bertini theorem for the case of a hy-persurface. Now it is possible to attract the new ideas related to irreducibility andtransversality of intersections of algebraic varieties. Quite probably (one shouldcheck the details) that in the case of the ground field of zero–characteristic onecan solve the problem (*) with the help of [2] (and without using methods of realalgebraic geometry).

These techniques are not sufficient for the case of the ground field of nonzerocharacteristic. Here the main difficulties are related to inseparability. But the situa-tion is not so hopeless. In the case of nonzero characteristic one can use additionallythe results [3]. We would like to formulate the following hypothesis.

(†) In the case on nonzero characteristic one can one construct a determinis-tic algorithm for computing the dimension of a projective algebraic varietyZ(f1, . . . , fm) with bitwise complexity polynomial in C(n)dn and the size ofthe input data where the constant C(n) depends only on n (more precisely,

C(n) < 22nC

for an absolute constant C > 0, cf. [3]).

37

A New Approach 3

References

[1] A.L. Chistov, Polynomial-time computation of the dimension of algebraic varieties in

zero-characteristic, Journal of Symbolic Computation. 1996, v.22, # 1, p. 1–25.

[2] A.L. Chistov, A bound for the degree of a system of equations determining the variety

of reducible polynomials. Algebra i Analiz , v. 24 (2012), No. 3, p. 199-222 (in Russian)[English translation: St.Petersburg Math. J., v.24 (2013), # 3, p. 513–528].

[3] A.L. Chistov, Efficient construction of local parameters for irreducible components of

an algebraic variety in nonzero characteristic, Zapiski Nauchnyh Seminarov POMIv. 326 (2005) p. 248-278 (in Russian) [English translation: Journal of MathematicalSciences v.140 (2007), # 3, p. 480–496].

[4] D.Lazard Résolution des systémes d’équations algébriques, Theor. Comput. Sci. v.15(1981), p. 77–110.

[5] F.S. Macauley, The algebraic theory of modular systems, Cambridge University Press,1916.

Alexander L. ChistovSt. Petersburg Department of Steklov Mathematical Instituteof the Academy of Sciences of RussiaFontanka 27, St. Petersburg 191023, Russiae-mail: [email protected]

38

Computer assisted constructive tasks with infinite

set of solutions for mathematical olympiads and

contests

Chukhnov, A. S., Posov, I. A.; and Pozdniakov S. N.

Abstract. The report presents a usage experience of constructive educationaltasks based on computer models. It is shown, that participants of competitionsmay construct many and various different solutions if they use software toolsbased on a computer model of a subject field to manipulate its objects. Asolution representation in terms of some construction allows for assessing thissolution by means of a set of formal criteria. Some criteria may be specifiedexplicitly as objective functions to be optimized by participants, others may bestated a posteriori to test different methodological hypothesis about solutionsfeatures.

From the point of view of automatic assessment, this approach can betreated as a transition from multiple choice tests to tasks with an infinite setof solutions. To specify a way to automatically asses a constructive solution,a teacher does not need to describe a solution that he or she should know inadvance. He or she should rather specify a set of criteria that must hold fora solution. Criteria used to analyze a solution also allow for assessing partialsolutions and providing feedback for participants while they work with a taskand thus adjust their work.

Authors also explore a usage of constructive tasks uas an intermediatestep to generalize partial solutions and ideas to justify the full solution. Theseries of competitions in discrete mathematics have been designed and im-plemented. This competitions suppose a constructive activity with softwaretools to be followed by theoretical tasks. Such series of tasks were also triedout as a part of the discrete mathematics course in a technical university.

During the experiments held inside the „Construct, Test, Explore” com-petition and inside the Olympiad in discrete mathematics and computer sci-ence, the constructive tasks proved to be appropriate for participant of dif-ferent level of preparation. But they also proved to have a drawback, thatparticipants overfocused on the experimental activity to the expense of theo-retical analysis of a task.

The work was supported by the Russian Foundation for Basic Research(Project No. 18-013-01130).

39

2

Chukhnov, A. S.Department of higher mathematics 2Saint Petersburg Electrotechnical University „LETI”Saint Petersburg, Russiae-mail: [email protected]

Posov, I. A.;Department of information systems in arts and humanitiesSaint Petersburg State UniversitySaint Petersburg, RussiaDepartment of higher mathematics 2Saint Petersburg Electrotechnical University „LETI”Saint Petersburg, Russiae-mail: [email protected]

Pozdniakov S. N.Department of higher mathematics 2Saint Petersburg Electrotechnical University „LETI”Saint Petersburg, Russiae-mail: [email protected]

40

Schutzenberger transformation on graded graphs:

Implementation and numerical experiments.

Vasilii Duzhin and Nikolay Vassiliev

1. Introduction

The Schutzenberger transformation on Young tableaux, also known as "jeu detaquin", was introduced in Schutzenberger’s paper [1]. This transformation allowsto solve different problems of enumerative combinatorics and representation theoryof symmetric groups. Particularly, it can be used to calculate the Littlewood-Richardson coefficients [2].

The connection between Schutzenberger transformation, RSK correspondence[3, 4, 5] and Markov Plancherel process [7] was found in [6]. The techniques dis-cussed in the work [6] have been developed in the recently published paper [8].

We consider the Schutzenberger transformation on two- and three- dimen-sional Young tableaux. The Schutzenberger transformation converts a Young tableauof size n to another Young tableau of size n− 1. At the beginning, the first box ofa source tableau is being removed. Then, the box with a smaller number is beingselected among top neighbouring and right neighbouring boxes. The selected box isthen being shifted to the position of the removed box. A newly formed empty boxis being filled by the neighbouring box using the same rule. This process continuesuntil the front of the diagram is reached.

The sequence of the shifted boxes forms so-called jeu de taquin path [8] orSchutzenberger path. Schutzenberger path is a path in Pascal graphs: Z2

+ or Z3+ in

2D and 3D cases, respectively.Besides the classic Schutzenberger transformation, in this work we also con-

sider two different modifications of it. In the first modification, we add an extra boxin the position of the last shifted box. In this case, the Schutzenberger transfor-mation does not change the shape of a diagram. Also the transformation becomesreversible, i.e. it establishes a bijection on the paths to a diagram. The secondmodification is a randomization of the classic Schutzenberger transformation. In

This work was supported by grant RFBR 17-01-00433.

41

2 Vasilii Duzhin and Nikolay Vassiliev

this case a path to a diagram on the third level of Young graph is being selectedrandomly. The results of numerical experiments suggest that the iterations of therandomized Schutzenberger transformation generate uniform distribution on thepaths to a diagram.

A. M. Vershik has noticed that the Schutzenberger algorithm can be appliednot only to the Young tableaux of an arbitrary dimension, but generally to anypartially ordered set. In this case the Schutzenberger transformation works onascendant sequences of decreasing ideals of a corresponding poset. Particularly,the technique of the Schutzenberger transformation can be used on any gradedgraph. In this situation, a Schutzenberger path will be a path on this gradedgraph.

It was proved in [8] that the Schutzenberger paths, obtained on two-dimensionalYoung tableaux, have a certain limit angle with a probability 1 relatively to thePlancherel measure.

Note that the standard Schutzenberger transformation is not reversible. Whereineach Young tableau has as many preimages as the number of transitions from agiven diagram of size n to the level n+ 1. Fig. 1 shows the Schutzenberger pathsof all preimages of the 2D Young tableau of size 106.

Figure 1. The Schutzenberger paths of all preimages of theYoung tableau of size 106.

1.1. The implementation of the Schutzenberger transformation

We propose the following algorithm for the implementation of the Schutzenbergertransformation on 2D and 3D Young tableaux. The same algorithm with minormodifications can be applied to any graded graph. We present Young tableaux as

42

Schutzenberger transformation on graded graphs. 3

arrays of sets of coordinates of added boxes. Note that the standard presentation ofYoung tableaux as two-dimensional arrays of integers has a significant disadvantagewith respect to the computational cost and memory usage. That is because inthat case the Schutzenberger transformation requires renumbering of all boxes ina tableau.

Let us consider the implemented algorithm for the case of 2D Young tableaux.During operation of the algorithm, the coordinates of boxes of a source tableau areprocessed consequently. At the beginning, the first box of a source tableau withcoordinates (0,0) is assigned as an active box. However, the active box is not beingadded to a new tableau immediately.

The active box is being added to a new tableau at the moment when aneighbour top or neighbour right box is added to a source tableau. As a next step,this neighbour box becomes active and so on. At the same time, the non-neighbourboxes are being added without any delay. The algorithm stops when all the boxesin a source tableau are processed.

Note that during operation of the algorithm, most of the boxes of a sourcetable are being copied to a new tableau without any changes. Only the order ofaddings of active boxes will be different. Another advantage of this approach isthat after necessary modifications it can be easily implemented on a Young graphof any dimension and on any other graded graphs.

We use the same methods to implement the modifications of Schutzenbergertransformation, i.e. the Schutzenberger transformation with the preservation ofshape of a diagram and with randomization. The fragment of the algorithm ofSchutzenberger transformation in 2D case, written in pseudocode, is shown below.Note that actX, actY are the coordinates of the current active box, in_tab is asource tableau and out_tab is a transformed tableau.

Listing 1. Schutzenberger transformation on 2D Young tableaux

1 actX = 0 ; actY = 0 ;2 f o r each (x , y ) from in_tab :3 4 i f ( ( x == actX + 1) && (y == actY ) ) | |5 ( ( x == actX ) && (y == actY + 1))6 7 out_tab . add ( actX , actY ) ;8 actX = x ; actY = y ;9 10 e l s e

11 12 out_tab . add (x , y ) ;13 14 15 out_tab . add ( actX , actY ) ;

43


2. Numerical experiments

Here we discuss the numerical experiments where the Schutzenberger transforma-tion was applied on large 2D and 3D Young tableaux. Particularly, we have gen-erated a 2D random Plancherel Young tableau of 3 million boxes. The Schutzen-berger transformation with the preservation of shape was consequently appliedto this tableau. The Vershik-Kerov coordinates x+y

√

3·106of the last boxes (x,y) of

Schutzenberger paths were recorded. The distribution of these coordinates is shownin Fig. 2.

Figure 2. The histogram of frequencies of last boxes of Schutzen-berger paths of 2D Young tableaux.

It can be seen from the figure that this histogram has the shape of so-calledsemicircle distribution. The same distribution was obtained in [9] as a limit dis-tribution of Plancherel probabilities on the front of large Young diagrams of sizen, n→∞. It has the following density function:

dµ(u) =

√4− u2

2 · π,

where u is one of Vershik-Kerov coordinates: u = x−y√n.

The next numerical experiment is devoted to the Schutzenberger transforma-tion on 3D Young graph. For each iteration of the Schutzenberger transformationwe compute the coordinates of the last boxes of Schutzenberger paths on the frontof random Young tableaux of a fixed shape. The distribution of the coordinates ob-tained in this experiment is shown in Fig. 3. Note that the size of the correspondingtableau is 3 million boxes.

As we can see, the distribution of last boxes of 3D Schutzenberger paths onthe front of the diagram is close to uniform. We plan to conduct more numericalexperiments to investigate this 3D distribution more precisely.

44

Schutzenberger transformation on graded graphs. 5

Figure 3. The distribution of coordinates of last boxes ofSchutzenberger paths of 3D Young tableaux.

Also we used the randomized Schutzenberger transformation to calculate theratio of dimensions of a pair of three-dimensional Young diagrams of sizes n andn + 1 which differ in a single box, i. e. a pair of diagrams connected with anedge in the Young graph. The co-transition probabilities of 3D central processescan be obtained using such ratios. The Schutzenberger transformation gives usthese co-transition probabilities without calculating the exact dimensions. That isespecially useful because there are no known three-dimensional analog of the 2Dhook length formula.

References

[1] M. P. Schutzenberger, "Quelques remarques sur une construction de Schensted",Math. Scandinavica 12, (1963), 117-128.

[2] S.V.Fomin, Knuth equivalence, jeu de taquin, and the Littlewood-Richardson rule,Appendix 1 to Chapter 7 in: R.P.Stanley, Enumerative Combinatorics, vol 2, Cam-bridge University Press.

[3] G. de B. Robinson, "On the representations of the symmetric group", American

Journal of Math. 60, (1938), 745-760.

[4] C. Schensted, "Longest increasing and decreasing subsequences", Canadian Journal

of Math. 13, (1961), 179-191.

[5] Donald E. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific

J. Math. Volume 34, Number 3 (1970), pp. 709-727.

[6] Sergei V. Kerov and Anatol M. Vershik. The characters of the infinite symmetricgroup and probability properties of the Robinson-Schensted-Knuth algorithm. SIAM

J. Algebraic Discrete Methods, 7(1):116–124, 1986

45


[7] S.V.Kerov and A.M.Vershik. Asymptotics of the Plancherel measure of the symmet-ric group and the limiting form of Young tableaux. Dokl. Akad. Nauk SSSR 233,No.6, 1024-1027 (1977).English translation: Sov. Math. Dokl. 18, 527-531 (1977).

[8] Dan Romik and Piotr Sniady. Jeu de taquin dynamics on infinite Young tableauxand second class particles. Annals of Probability: An Official Journal of the Institute

of Mathematical Statistics, 43(2):682-737, 2015

[9] S. V. Kerov, “Transition Probabilities for Continual Young Diagrams and the MarkovMoment Problem”, Funktsional. Anal. i Prilozhen., 27:2 (1993), 32–49; Funct. Anal.

Appl., 27:2 (1993), 104–117

Vasilii DuzhinSaint Petersburg Electrotechnical Universityul. Professora Popova 5, 197376 St. Petersburg, Russian Federatione-mail: [email protected]

Nikolay VassilievSt.Petersburg department of Steklov Institute of mathematics RASnab.Fontanki 27,St.Petersburg, 191023, Russiae-mail: [email protected]

46

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+ 84x4y − 24√6a1x

3y2 − 36b0x3y2 + 21/3

(

x3

y2

)1/3

y2 ·(

√6 +

(

x3

y2

)2/3)

×

(2(

9a1 + 4√6b0

)

x3 + 3(

3√6a1 + 8b0

)

y2)

×

2F1

(

−1/2, 1/3; 1/2; 3y2

3y2+√

6x3

) ]

,

I6(x, y) = y ·(

√

2/3 +x3

y2

)−12+

a1

−6a1−2√

6b0

(

x2

y

)−a1

3a1+√

6b0

×

3 +x2

y2[√

6x+ 3(

2a1 +√6b0

)

y]

.

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+* 61 1++ #1 *4+ 1/1*+3

dx

dt= y + 2xy,

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dt= − x− bx2 + cxy + y2.

=>?

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x = 0, y = 0,x = −(1/b), y = 0,

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C* b = 1; I(x, y) = (1 + 2x)(1

4(−4 + c(c+

√c2 − 4)),

C* c = 0; I(x, y) = (−1− 2bx(1 + x)− 2y2 + (b− 1)(2x+ 1) log 2x+ 1)/(8x+ 4).=E?

)% (!*4 "#1+1 :+ 4#5+ *4+ .%5#-.#%* $.%+ x = −1/2 :4."4 1+7#-#*+1 *4+ $+2* 7#-*D:4+-+ +8.1*1 *4+ ,$!(#$ B-1* .%*+,-#$ ="+%*+- "#1+? 2-!3 -.,4* 7#-* 1.&+ :4+-+ .%*+F

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49

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]-22 &**G Q*/I$+ F>6 *'G+ V*'*)1*+ R*)A*)+ 9!7XL; I L!Z$

50

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!"#$% &'()%*+

,-$.)+#/0( 1(/#!#2#) $3 42"+)*% 560/!"/7

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)J9*!+N 2$% #*+#)/$")2,4*&0"-5/6+,-

51

Thomas decomposition of differential systems and

its implementation in Maple

Vladimir Gerdt, Markus Lange-Hegermann and Daniel Robertz

We present the basic algorithmic features and implementation in Maple of

the differential Thomas decomposition of polynomially nonlinear differential sys-

tems, which in addition to equations may contain inequations, into a finite set

of differentially triangular and algebraically simple subsystems whose subsets of

equations are involutive. Usually the decomposed system is substantially easier to

investigate and solve both analytically and numerically. The distinctive property

of a Thomas decomposition is disjointness of the solution sets of the output sub-

systems. Thereby, a solution of a well-posed initial problem belongs to one and

only one output subsystem. The Thomas decomposition is fully algorithmic. It

allows to perform important elements of algebraic analysis of an input differen-

tial system such as: verifying consistency, i.e., the existence of solutions; detecting

the arbitrariness in the general analytic solution; given an additional equation,

checking whether this equation is satisfied by all common solutions of the input

system; eliminating a part of dependent variables from the system if such elimina-

tion is possible; revealing hidden constraints on dependent variables, etc. Examples

illustrating the use of the differential Thomas decomposition are given.

References

[1] Thomas Bächler, Vladimir Gerdt, Markus Lange-Hegermann and Daniel Robertz.Algorithmic Thomas decomposition of algebraic and differential systems. Journal ofSymbolic Computation 47(10), 1233–1266, 2012.

[2] Daniel Robertz. Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathe-matics, Vol. 2121, Springer, 2014.

[3] Vladimir Gerdt, Markus Lange-Hegermann and Daniel Robertz. The MAPLE pack-

age TDDS for computing Thomas decompositions of systems of nonlinear PDEs.arXiv:1801.09942 [physics.comp-ph]

52

2 Vladimir Gerdt, Markus Lange-Hegermann and Daniel Robertz

Vladimir GerdtLaboratory of Information TechnologiesJoint Institute for Nuclear Research141980 Dubna, Russiae-mail: [email protected]

Markus Lange-HegermannUniversity of Applied Science Ostwestfalen-Lippe32656 Lemgo, Germanye-mail: [email protected]

Daniel RobertzSchool of Computing, Electronics and MathematicsPlymouth UniversityPL4 8AA Plymouth, United Kingdome-mail: [email protected]

53

Upper bounds on Betti numbers of tropical pre-

varieties

Dima Grigoriev and Nicolai Vorobjov

We prove upper bounds on the sum of Betti numbers of tropical prevarieties

in dense and sparse settings. In the dense setting the bound is in terms of the

volume of Minkowski sum of Newton polytopes of defining tropical polynomials,

or, alternatively, via the maximal degree of these polynomials. In sparse setting,

the bound involves the number of the monomials.

Dima Grigoriev

French National Center for Scientific Research

3, rue Michel-Ange

75794 Paris cedex 16

France

e-mail: [email protected]

Nicolai Vorobjov

University of Bath

Claverton Down

Bath

BA2 7AY

United Kingdom


54

Double Hurwitz Numbers

Maksim Karev

Abstract. The talk is based on the joint work with N. Do (Monash Univer-sity). The most straightforward definition of the double Hurwitz numbersDg(µ, ν) is, up to a multiplicative constant, the number of ways to multi-ply a given permutation of cyclic type µ by a product of 2g − 2 + |µ| + |ν|transpositions such that the result is of cyclic type ν. It turns out that thesenumber can be packed into generating functions that can be calculated usinga recursion. We formulate a conjecture on the analytical properties of thesegenerating functions.

Introduction

The talk is based on the joint work with N. Do (Monash University). SimpleHurwitz numbers enumerate the number of ways to decompose a permutation ofa given cyclic type into a product of fixed number of transpositions. Their studywas first initiated in nineteenth century by A. Hurwitz. However, they still attractthe interest due to incredibly rich structure they possess.

Double Hurwitz numbers are defined in a similar way: we fix two cyclic typeµ and ν in the symmetric group Sd and count the number of ways to multiply apermutation of a cyclic type µ by a product of 2g − 2 + |µ| + |ν| transpositionssuch that the result is of cyclic type ν.

It is well-known that both simple and double Hurwitz numbers can be in-terpreted as a number of non-isomorphic ramified covers of CP 1 with certainrestriction on the branch points profiles. It allows us to compute double Hurwitznumbers via the enumeration of ramified covers weighted by a certain polynomialweight as follows.

Fix a positive integer d and weights s, q1, q2, . . . , qd ∈ C. Define the double

Hurwitz number DHg,n(µ1, . . . , µn) to be the weighted count of connected genus

g branched covers of the Riemann sphere f : (Σ; p1, . . . , pn)→ (CP1;∞) such that

• all branching away from 0 and ∞ is simple and occurs at some number m offixed points;

55

2 Maksim Karev

• f−1(∞) = µ1p1 + · · ·+ µnpn; and• no preimage of 0 has ramification index larger than d.

If such a branched cover has ramification profile (λ1, λ2, . . . , λℓ) over 0, then weassign it the weight

qλ1qλ2· · · qλℓ

|Aut f |

sm

m!.

Here, the automorphism group Aut f consists of Riemann surface automorphismsφ : Σ→ Σ that preserve the marked points p1, . . . , pn and satisfy f φ = f .

We present an efficient recursion that, in principle, allows to compute all dou-ble Hurwitz numbers, and formulate an explicit conjecture concerning the proper-ties of the corresponding generating functions.

Maksim KarevRepresentation theory and dynamical systems laboratoryPDMI RASSt Petersburg, Russiae-mail: [email protected]

56

Irreducible Decomposition of Representations of

Finite Groups via Polynomial Computer Algebra

Vladimir V. Kornyak

Abstract. An algorithm for splitting permutation representations of finitegroup over fields of characteristic zero into irreducible components is de-scribed. The algorithm is based on the fact that the components of the in-variant inner product in invariant subspaces are operators of projection intothese subspaces. An important part of the algorithm is the solution of sys-tems of quadratic equations. A preliminary implementation of the algorithmsplits representations up to hundreds of thousands of dimensions. Examplesof computations are given.

1. Introduction. One of the central problems of group theory and its applicationsin physics is the decomposition of linear representations of groups into irreduciblecomponents. In general, the problem of splitting a module over an associative alge-bra into irreducible submodules is quite nontrivial. An overview of the algorithmicaspects of this problem can be found in [1]. For vector spaces over finite fields,the most efficient is the Las Vegas type algorithm called MeatAxe. This algorithmplayed an important role in solving the problem of classifying finite simple groups.However, the approach used in the MeatAxe is ineffective in characteristic zero,whereas quantum-mechanical problems are formulated just in Hilbert spaces overfields of characteristic zero. Our algorithm deals with representations over suchfields, and its implementation copes with dimensions up to hundreds of thousandsthat is not less than the dimensions achievable for the MeatAxe. The algorithmrequires knowledge of the centralizer ring of the considered group representation.In the general case, the calculation of the centralizer ring is a problem of linearalgebra, namely, solving matrix equations of the form AX = XA. In the case ofpermutation representations, there is an efficient algorithm for computing the cen-tralizer ring — it is reduced to constructing the set of orbitals. In addition, permu-tation representations are fundamental in the sense that any linear representationof a finite group is a subrepresentation of some permutation representation, andwe use this fact in some quantum mechanical considerations [2, 3]. Therefore, weconsider here only permutation representations.2. Mathematical preliminaries. Let G be a transitive permutation group on the setΩ ∼= 1, . . . ,N. The action of g ∈ G on i ∈ Ω is denoted by ig. A representationof G in an N-dimensional vector space over a field F by the matrices P(g) withthe entries P(g)ij = δigj , where δij is the Kronecker delta, is called a permutation

representation. We assume that the permutation representation space is a Hilbertspace HN. From a constructive point of view it is sufficient to assume that the

57

2 Vladimir V. Kornyak

base field F is a minimal splitting field of the group G. Such field is a subfield ofan m-th cyclotomic field, where m is a divisor of the exponent of G. The field F ,being an abelian extension of Q, is a constructive dense subfield of R or C.

An orbit of G on the Cartesian square Ω × Ω is called an orbital [5]. Thenumber of orbitals, R, is called the rank of G on Ω. Among the orbitals of a transi-tive group there is one diagonal orbital, ∆1 = (i, i) | i ∈ Ω, which will always befixed as the first element in the list of orbitals ∆1, . . . ,∆R. For a transitive actionof G there is a natural one-to-one correspondence between the orbitals of G andthe orbits of a point stabilizer Gi: ∆←→ Σi = j ∈ Ω | (i, j) ∈ ∆ . The Gi-orbitsare called suborbits and their cardinalities are called the suborbit lengths.

The invariance condition for a bilinear form A in the Hilbert space HN canbe written as the system of equations A = P(g)AP

(g−1

), g ∈ G. It is easy to

verify that in terms of the entries the equations of this system have the form(A)ij = (A)igjg . Thus, the matrices A1, . . . ,AR, where Ar is the characteris-

tic function of the orbital ∆r on the set Ω × Ω, form a basis of the centralizer

ring of the representation P. The multiplication table for this basis has the form

ApAq =∑R

r=1Cr

pqAr, where Crpq are non-negative integers. The commutativity of

the centralizer ring indicates that the representation P is multiplicity-free.3. Algorithm and its implementation. Let T be a transformation (we can assumethat T is unitary) that splits the permutation representation P into M irreduciblecomponents:

T−1P(g)T = 1⊕ Ud2(g)⊕ · · · ⊕ Udm

(g)⊕ · · · ⊕ UdM(g) ,

where Udmis a dm-dimensional irreducible subrepresentation, ⊕ denotes the direct

sum of matrices, i.e., A⊕B = diag(A,B).The matrix 1N is the standard inner product in any orthonormal basis. In the

splitting basis we have the following decomposition of the standard inner product

1N = 1d1=1⊕ · · · ⊕ 1dm⊕ · · · ⊕ 1dM

.

The inverse image of this decomposition in the original permutation basis is

1N = B1 + · · ·+ Bm + · · ·+ BM ,

where Bm is defined by

T−1BmT = 01+d2+···+dm−1⊕1dm

⊕ 0dm+1+···+dM.

The main idea of the algorithm is based on the fact that Bm’s form a completeset of orthogonal projectors, i.e., they are idempotent, B2m = Bm, and mutuallyorthogonal, BmBm′ = 0N if m 6= m′. We see that all Bm’s can be obtained assolutions of the idempotency equation X2−X = 0N for the generic invariant formX = x1A1+ · · ·+xRAR. This is a system of quadratic polynomial equations in theindeterminates x1, x2, . . . , xR. The polynomial system can be computed by usingthe multiplication table. Let us write the projector in the basis of invariant forms:Bm = bm,1A1+bm,2A2+ · · ·+bm,RAR. It is easy to show that bm,1 = dm/N. Thus,any solution of the idempotency system has the form [x∗

1 = d/N, x∗

2, . . . , x∗

R] , whered ∈ [1..N− 1] is either an irreducible dimension or a sum of such dimensions.

58

Irreducible Decomposition of Representations of Finite Groups 3

The core part of the algorithm is constructed as follows.

We set initially E(x1, x2, . . . , xR)←X2 −X = 0N

.

Then we perform a loop on dimensions that starts with d = 1 and ends whenthe sum of irreducible dimensions becomes equal to N.

For the current d we solve the system of equations E(d/N, x2, . . . , xR) . All so-lutions belong to abelian extensions of Q, so their getting is always algorithmicallyrealizable.

If the system is incompatible, then go to the next d.

If E(d/N, x2, . . . , xR) describes a zero-dimensional ideal, then we have k (in-cluding the case k = 1) different d-dimensional irreducible subrepresentations.

If the polynomial ideal has dimension h > 0, then we encounter an irreduciblecomponent with a multiplicity k, where

⌊k2/2

⌋= h. In this case we select, by a

somewhat arbitrary procedure, k convenient mutually orthogonal representativesin the family of equivalent subrepresentations.

In any case, if at the moment we have a solution Bm, we append Bm tothe list of irreducible projectors, and exclude from the further consideration thecorresponding invariant subspace by adding the linear orthogonality condition

BmX = 0N to the polynomial system:

E(x1, x2, . . . , xR)← E(x1, x2, . . . , xR) ∪ BmX = 0N .

After processing all Bm’s of dimension d, go to the next d.

The complete algorithm is implemented by two procedures:

1. The procedure PreparePolynomialData is a program written in C. The in-put data for this program is a set of permutations of Ω that generates thegroup G. The program computes the basis of the centralizer ring and itsmultiplication table, constructs the idempotency and orthogonality polyno-mials, and generates the code of the procedure SplitRepresentation thatprocesses the polynomial data. The implementation is able to cope with di-mensions (dimension= |Ω|) up to several hundred thousand on a PC withina reasonable time.

2. The procedure SplitRepresentation implements the above described loopon dimensions that splits the representation of the group into irreduciblecomponents. It is generated by the C program PreparePolynomialData.Currently, the code is generated in the Maple language, and the polyno-mial equations are processed by the Maple implementation of the Gröbnerbases algorithms.

Comparison with the Magma implementation of the MeatAxe.

The Magma database contains a 3906-dimensional representation of the ex-ceptional group of Lie type G2(5). This representation (over the field GF(2)) isused in [4] as an illustration of the capabilities of the MeatAxe.

The application of our algorithm to this problem — the calculation showedthat the splitting field in this case is Q — produces the following data.

59

4 Vladimir V. Kornyak

Rank: 4. Suborbit lengths: 1, 30, 750, 3125.

3906 ∼= 1⊕ 930⊕ 1085⊕ 1890

B1 =1

3906

4∑

k=1

Ak

B930 =5

21

(

A1 +3

10A2 +

1

50A3 −

1

125A4

)

B1085 =5

18

(

A1 −1

5A2 +

1

25A3 −

1

125A4

)

B1890 =15

31

(

A1 −1

30A2 −

1

30A3 +

1

125A4

)

Time C: 1.14 sec. Time Maple: 0.8 sec.The Magma fails to split the 3906-dimensional representation over the field Q,but we can model to some extent the case of characteristic zero, using a field ofcharacteristic not dividing |G2(5)|. The smallest such field is GF(11).

Below is the session of the correspondingMagma computation on a computerwith two Intel Xeon E5410 2.33GHz CPUs (time is given in seconds).

> load "g25";

Loading "/opt/magma.21-1/libs/pergps/g25"

The Lie group G( 2, 5 ) represented as a permutation

group of degree 3906.

Order: 5 859 000 000 = 2^6 * 3^3 * 5^6 * 7 * 31.

Group: G

> time Constituents(PermutationModule(G,GF(11)));

[

GModule of dimension 1 over GF(11),



GModule of dimension 1890 over GF(11)

]

Time: 282.060

4. Some decompositions for sporadic simple groups.

Generators of representations are taken from the section “Sporadic groups”of the Atlas [6].

Representations are denoted by their dimensions in bold (possibly with somesigns added to distinguish different representations of the same dimension).

Permutation representations are underlined.

Multiple subrepresentations are underbraced in the decompositions.

All timing data were obtained on a PC with 3.30GHz Intel Core i3 2120 CPU.

60

Irreducible Decomposition of Representations of Finite Groups 5

• 1980-dimensional representation of the Mathieu group cover 6.M22

Rank: 17. Suborbit lengths: 16, 143, 843, 3365.

1980 ∼= 1⊕ 21α ⊕ 21β ⊕ 21β ⊕ 55⊕ 99α ⊕ 99β ⊕ 99β ⊕ 105+ ⊕ 105+

⊕ 105− ⊕ 105− ⊕ 120⊕ 154⊕ 210⊕ 330⊕ 330

Time C: 2 sec. Time Maple: 8 h 41 min 1 sec.• 29155-dimensional representation of the Held group HeRank:12.Suborbit lengths: 1, 90, 120, 384, 9602, 1440, 2160, 28802, 5760, 11520.

29155 ∼= 1⊕ 51⊕ 51⊕ 680⊕ 1275⊕ 1275︸︷︷︸

⊕1920⊕ 4352

⊕ 7650⊕ 11900

Time C: 5 min 41 sec. Time Maple: 15 sec.• 66825-dimensional representation of the McLaughlin group cover 3.McLRank: 14. Suborbit lengths: 13, 630, 22403, 50403, 80643, 20160.

66825 ∼= 1⊕ 252⊕ 1750⊕ 2772⊕ 2772⊕ 5103α ⊕ 5103β ⊕ 5103β

⊕ 5544⊕ 6336⊕ 6336⊕ 8064⊕ 8064⊕ 9625

Time C: 39 min 36 sec. Time Maple: 14 min 11 sec.• 98280-dimensional representation of the Suzuki group cover 3.SuzRank: 14. Suborbit lengths: 13, 8913, 28163, 5940, 19008, 207363.

98280 ∼= 1⊕ 78⊕ 78⊕ 143⊕ 364⊕ 1365⊕ 1365⊕ 4290⊕ 4290

⊕ 5940⊕ 12012⊕ 14300⊕ 27027⊕ 27027

Time C: 2 h 36 min 29 sec. Time Maple: 7 min 41 sec.

References

[1] Holt, D. F., Eick, B., O’Brien, E. A. Handbook of Computational Group Theory.

Chapman & Hall/CRC, 2005.

[2] Kornyak V. V. Quantum models based on finite groups. J. Phys.: Conf. Ser. 965

012023, 2018. http://stacks.iop.org/1742-6596/965/i=1/a=012023

[3] Kornyak V. V. Modeling Quantum Behavior in the Framework of PermutationGroups. EPJ Web of Conferences 173 01007, 2018.https://doi.org/10.1051/epjconf/201817301007

[4] Bosma, W., Cannon, J., Playoust, C., Steel, A. Solving Problems with Magma.

University of Sydney. http://magma.maths.usyd.edu.au/magma/pdf/examples.pdf

[5] Cameron P. J. Permutation Groups. Cambridge University Press, 1999.

[6] Wilson, R. A., et al., Atlas of finite group representations.

http://brauer.maths.qmul.ac.uk/Atlas/v3.

Vladimir V. KornyakLaboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubna, Russiae-mail: [email protected]

61

On the Cayley-Bacharach Property

Martin Kreuzer, Le Ngoc Long and Lorenzo Robbiano

Abstract. The Cayley-Bacharach property, which has been classically stated

as a property of a finite set of points in an affine or projective space, is

extended to arbitrary 0-dimensional affine algebras over arbitrary base fields.

We present characterizations and explicit algorithms for checking the Cayley-

Bacharach property directly, via the canonical module, and in combination

with the property of being a locally Gorenstein ring. Moreover, we characterize

strict Gorenstein rings by the Cayley-Bacharach property and the symmetry

of their Hilbert function, as well as by the strict Cayley-Bacharach property

and the last difference of their Hilbert function.

Extended Abstract

The Cayley-Bacharach Property (CBP) has a long and rich history. Classically, ithas been formulated geometrically as follows: A set of points X in n-dimensionalaffine or projective space is said to have the Cayley-Bacharach property of degree dif any hypersurface of degree d which contains all points of X but one automat-ically contains the last point. After a brief recap of its history, we present thecurrently most general version, namely the definition first given in Ngoc Le Long’sThesis (University of Passau, 2015). Our goal is to study this very general versionof the CBP and to find efficient algorithms for checking it. A special emphasisis given to algorithms which will us to apply them to families of 0-dimensionalideals parametrized by border basis schemes. Moreover, we generalize the mainresults about the CBP of many previous papers to this most general setting of a0-dimensional affine algebra over an arbitrary base field.

To achieve these goals, we proceed as follows. Our main object of study isa 0-dimensional affine algebra R = P/I over an arbitrary field K, where we letP = K[x1, . . . , xn] be a polynomial ring over K and I a 0-dimensional ideal in P .Even if we do not specify it explicitly everywhere, we always consider R togetherwith this fixed presentation. In other words, we consider a fixed 0-dimensionalsubscheme X = Spec(P/I) of An.

62

2 Martin Kreuzer, Le Ngoc Long and Lorenzo Robbiano

This corresponds to the classical setup. However, in the last decades it hasbeen customary to consider 0-dimensional subschemes of projective spaces. Ofcourse, via the standard embedding An ∼= D+(x0) ⊂ P

n, the classical setup can betranslated to this setting in a straightforward way. For instance, in this case theaffine coordinate ring R = K[x1, . . . , xn]/I has to substituted by the homogeneouscoordinate ring Rhom = K[x0, . . . , xn]/I

hom, etc. In this talk we use the affinesetting for several reasons: firstly, the ideals defining subschemes of X can bestudied using the decomposition into local rings, secondly, the structure of thecoordinate ring of X and its canonical module can be described via multiplicationmatrices, and thirdly, the affine setup is suitable for generalizing everything tofamilies of 0-dimensional ideals via the border basis scheme.

First we recall the primary decomposition I = Q1 ∩ · · · ∩Qs of I, the corre-sponding primary decomposition 〈0〉 = q1 ∩ · · · ∩ qs of the zero ideal of R, and thedecomposition R = R/q1×· · ·×R/qs of R into local rings. Then, for i ∈ 1, . . . , s,a minimal Qi-divisor J of I is defined in such a way that the corresponding sub-scheme of X differs from X only at the point pi = Z(Mi) and has the minimalpossible colength ℓi = dimK(P/Mi), where Mi = Rad(Qi). In the reduced case,these subschemes are precisely the sets X \ pi appearing in the classical formu-lation of the Cayley-Bacharach Theorem.

Moreover, in order to have a suitable version of degrees, we recall the degreefiltration of R, its affine Hilbert function HFa

R, and its regularity index ri(R). Here

the affine Hilbert function plays the role of the usual Hilbert function if we consideraffine algebras such as R.

These constructions are combined with the definition and some characteri-zations of separators. Then we show that a separator for a maximal ideal mi of Rcorresponds to a generator of a minimal Qi-divisor J of I, and we use the maximalorder of such a separator to describe the regularity index of J/I. Then the min-imum of all regularity indices ri(J/I) is called the separator degree of mi. We goon to show that this “minimum of all maxima” definition is the correct, but rathersubtle generalization of the classical notion of the least degree of a hypersurfacecontaining all points of X but pi.

The separator degree of a maximal ideal mi of R is bounded by the regularityindex ri(R), since the order of any separator is bounded by this number. If allseparator degrees attain this maximum value, we say that R has the Cayley-Bacharach property (CBP), or that X is a Cayley-Bacharach scheme. At this pointwe construct our first new algorithm which allows us to check whether a givenmaximal ideal mi of R has maximal separator degree.

Although this algorithm can be used to check the CBP of R, we then constructa better one based on the canonical module ωR = HomK(R,K) of R. The modulestructure of ωR is given by (f ϕ)(g) = ϕ(fg) for all f, g ∈ R and all ϕ ∈ ωR.It carries a degree filtration G = (GiωR)i∈Z which is given by GiωR = ϕ ∈ωR | ϕ(F−i−1R) = 0 and its affine Hilbert function which satisfies HFa

ωR(i) =

dimK(R)−HFa

R(−i− 1) for i ∈ Z. Generalizing some earlier results, we show that

the module structure of ωR is connected to the CBP of R. More precisely, one

63

On the Cayley-Bacharach Property 3

main theorem says that R has the CBP if and only if AnnR(G− ri(R)ωR) = 0.Based on this characterization and the description of the structure of R and themodule structure of ωR via multiplication matrices, we obtain the second mainalgorithm for checking the CBP of R using the canonical module. As a nice anduseful by-product, we show that, for an extension field L of K, the ring R hasthe CBP if and only if R⊗K L has the CBP.

Next we turn our attention to 0-dimensional affine algebras R which arelocally Gorenstein and have the CBP. We show that R is locally Gorenstein if andonly if ωR contains an element ϕ such that AnnR(ϕ) = 0 and that we can checkthis effectively. Then we characterize locally Gorenstein rings having the CBP bythe existence of an element ϕ ∈ ωR⊗L of order − ri(R) with AnnR⊗L(ϕ) = 0.Here we may have to use a base field extension K ⊆ L or assume that K is infinite.This characterization implies useful inequalities for the affine Hilbert function of Rand allows us to formulate an algorithm which checks whether R is a locallyGorenstein ring having the CBP using the multiplication matrices of R. To endthis discussion, we characterize the CBP of R in the case when the last difference∆R = HFR(ri(R))−HFR(ri(R)− 1) is one.

The subsequent topic is to characterize 0-dimensional affine algebras whichare strict Gorenstein rings. This property means that the graded ring grF(R) withrespect to the degree filtration is a Gorenstein ring. In the projective case, thecorresponding 0-dimensional schemes are commonly called arithmetically Goren-stein. Our first characterization of strict Gorenstein rings improves earlier resultsby Davis, Geramita, and Orecchia. More precisely, we show that R is strictlyGorenstein if and only if it has the CBP and a symmetric Hilbert function. Inparticular, it follows that these rings are locally Gorenstein. Then we define thestrict CBP of R by the CBP of grF (R) and show that it implies the CBP of R.Thus we obtain a second characterization of strict Gorenstein rings: R is a strictGorenstein ring if and only if R has the strict CBP and ∆R = 1.

In the last part of the talk, we show how one can extend all these char-acterizations to families of 0-dimensional polynomial ideals. More precisely, weintroduce the border basis scheme and explain some ways of getting explicit poly-nomial equations defining subschemes corresponding to all ideals with a particularproperty, for instance the CBP.

Martin Kreuzer

Fakultät für Informatik und Mathematik, Universität Passau, D-94030 Passau, Germany


Le Ngoc Long

Fakultät für Informatik und Mathematik, Universität Passau, D-94030 Passau, Germany


64

4 Martin Kreuzer, Le Ngoc Long and Lorenzo Robbiano

Lorenzo Robbiano

Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova,

Italy


65

余英

F (x, y, y′, . . . ) = 0

x

y′ = f(x, y), f ∈ Q(x, y),

66

余英

~x = f(~x),

ρ ~x

ρ > 1

ρ = 1

67

余英

68

What does a random knot look like?

Andrei Malyutin

Abstract. We discuss the structure and statistics of the set of classical knotsand present new results in this research area.

We study the structure and statistical characteristics of the set of classicalknots. Closely related topics are statistics of links, tangles, 3-manifolds, graphembeddings, plane diagrams, meanders, countable groups, elements of mappingclass groups, braids, etc. The question we primarily address is what does a typicallarge (prime) knot look like. Probabilistic wording for this question is as follows:what properties does a random (prime) knot have?

What properties of knots does it make sense to check for genericity? Nicecandidates come from the basic knot classification. The first level of the classifica-tion splits the set of non-trivial knots into the subclasses of prime and compositeones. The second level divides prime knots into the satellite (with incompressibletori in the complement) and simple ones. Similar classifications hold for links, 3-manifolds, etc. Thurston proved that the complement of every simple knot bearsa geometric structure: every simple knot is either torus or hyperbolic. This yieldsthe following tree of knots basic properties:

KNOTS

UNKNOT

NON-TRIVIAL

COMPOSITE

PRIME

SATELLITE

SIMPLE

TORUS

HYPERBOLIC

Figure 1. The tree of basic knot classes.

Supported by RFBR grant 16-01-00609.

69

2 Andrei Malyutin

The dual question to the previous one is which random knot model we choose.Dozens of such models are described in the literature (see [4]). We discuss severalof them that are either natural, well-studied, or have specific properties that areof particular interest to our discussion. Our list includes the following:

• random walk models (a random knot is a random polygon in R3 whose edge

vectors are guided by some non-degenerate probability distribution);• braid group models (we consider knots and links that are Alexander/plat

closures of randomly generated braids);• knot tables model (we consider uniform measures on the sets of knots with

crossing number at most n);• random jump model (a random knot is a random polygon in R

3 whose verticesare guided by the same non-degenerate probability distribution).

An interesting issue related to the above properties of knots is the balancebetween hyperbolic knots and satellites. For a rather long period of time it waswidely believed that most knots and links are hyperbolic (see, e. g., [12, p. 507]).The reason is that the sets of torus and satellite knots look rather special andrare and give an impression of scarcity. In particular, only 32 of the first 1 701 936prime knots are non-hyperbolic (see [5]). Another related fact is that hyperbolicknots are generic in the braid group models (see [9, 7, 6]).

However, a deeper analysis shows that the conjecture of the hyperbolic knotsprevalence is quite flimsy. Indeed, the braid group models are highly imbalanced,and the case with 1 701 936 knots can be explained by the fact that satellites arerelatively large, which does not imply asymptotic scarcity. Furthermore, there is(indirect) evidence that the satellites persist in random walk models (see [8, 3]). Inaddition, the conjecture that hyperbolic knots are asymptotically generic in primeknot tables (see [1]) contradicts several other plausible conjectures (see [10, 11]).This is due to the fact that satellite structures should be large enough, but theycan be local (see [11]). We also present the following new evidence related to thetables model.

Theorem 1. The percentage of hyperbolic links amongst all of the prime non-split

links of n or fewer crossings does not tend to 100 as n tends to infinity.

In models where a random knot is satellite, it is interesting to study itscompanionship tree (see [2]).

It would seem that the above arguments indicate satellite knots predom-inance. However, we conjecture that the space of knots is complex enough tohave several natural well-balanced random knot models showing opposite behav-ior. In this regard, we present the following new conjecture related to the randomjump model.

Conjecture 1. Hyperbolic knots are generic in the random jump model.

Expected behavior of distinct random knot models is presented in Table 1.

70

What does a random knot look like? 3

model \ set of knots all knots prime knotsrandom walk models composite (proved) satellite (str. conj.)braid group models hyperbolic (proved) hyperbolic (proved)knot tables model composite (str. conj.) satellite (str. conj.)

random jump model hyperbolic (weak conj.) hyperbolic (weak conj.)

Table 1. Generic types of knots in several models.

References

[1] C.C. Adams, The Knot Book: An Elementary Introduction to the Mathematical The-

ory of Knots, New York, W. H. Freeman and Company, 1994.

[2] R. Budney, JSJ-decompositions of knot and link complements in S3, Enseign. Math.

(2) 52:2 (2006), 319–359.

[3] Y. Diao, J. C. Nardo, and Y. Sun, Global knotting in equilateral random polygons,J. Knot Theory Ramifications 10:4 (2001), 597–607.

[4] C. Even-Zohar, Models of random knots, J. Appl. Comput. Topology 1:2 (2017),263–296.

[5] J. Hoste, M. Thistlethwaite, and J. Weeks, The first 1,701,936 knots, Math. Intelli-gencer 20:4 (1998), 33–48.

[6] K. Ichihara and J. Ma, A random link via bridge position is hyperbolic, TopologyAppl. 230 (2017), 131–138.

[7] T. Ito, On a structure of random open books and closed braids, Proc. Japan Acad.Ser. A Math. Sci. 91:10 (2015), 160–162.

[8] D. Jungreis, Gaussian random polygons are globally knotted, J. Knot Theory Rami-fications 3:4 (1994), 455–464.

[9] J. Ma, The closure of a random braid is a hyperbolic link, Proc. Amer. Math. Soc.142:2 (2014), 695–701.

[10] A.V. Malyutin, Satellite knots strike back, Abstracts of the International Conference“Polynomial Computer Algebra ’15” (2015), 65–66.

[11] A.V. Malyutin, On the question of genericity of hyperbolic knots, preprint (2016),arXiv:1612.03368.

[12] J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Grad. Texts in Math. 149,Springer, New York, 2nd edition, 2006.

Andrei MalyutinSt. Petersburg Department ofSteklov Institute of MathematicsSt. Petersburg State UniversitySt. Petersburg, Russiae-mail: [email protected]

71

On Strongly Consistent Finite Difference Approx-

imations

Dominik Michels, Vladimir Gerdt, Dmitry Lyakhov and Yuri Blinkov

Solving partial differential equations (PDEs) belongs to the most fundamentaland practically important research challenges in mathematics and in the compu-tational sciences. Such equations are typically solved numerically since obtainingtheir explicit solution is usually very difficult in practice or even impossible. Oneof the classical and nowadays well-established and popular approaches is the finitedifference method [1, 2, 3] which exploits a local Taylor expansion to replace adifferential equation by the difference one. This raises the question how to pre-serve fundamental properties of the underlying PDEs at the discrete level. From ageometric point of view, the most important properties are symmetries and con-servation laws. Importance of conservation laws in mathematical physics could notbe underestimated, since many fundamental properties for nonlinear PDEs (likeexistence and uniqueness of solutions) typically are based on conservation laws.From algebraic perspective, the basic object which should be preserved is algebraicrelations between equations and their differential (difference) consequences. Theproblem here occurs because finite difference approximation of derivation doesn’tsatisfy Leibnitz rule.

The fundamental requirement of a finite difference scheme (FDS) is its con-vergence to a solution of the corresponding differential problem as the grid spac-ings go to zero. According to the Lax-Richtmyer equivalence theorem [4, 5], fora scalar PDE it has been adopted that the convergence is provided if a givenfinite-difference approximation (FDA) to the PDE is consistent and stable. Theconsistency implies a reduction of the FDA to the original PDE when the gridspacings go to zero, and it is obvious that the consistency is necessary for con-vergence. The theorem states that a FDS for an initial value (Cauchy) problemproviding the existence and uniqueness of the solution converges if and only if itsFDA is consistent and numerically stable.

In this talk we describe algorithmic methods to generate FDAs to PDEs onorthogonal and uniform grids, and to verify strong consistency of the obtainedFDAs. The main algorithmic tool for the case of linear PDEs is the differenceelimination provided by Groebner bases [6, 7, 8] for a certain elimination ranking.

72

2 Dominik Michels, Vladimir Gerdt, Dmitry Lyakhov and Yuri Blinkov

Figure 1. Simulation of the Kármán vortex street computedwith the new FDA. The characteristic repeating pattern ofswirling vortices can be observed, cf. [15].

Given a system of polynomially-nonlinear PDEs and its FDA, the s-consistencyanalysis is based on a computation of a difference standard Groebner basis and theconstruction of a differential Thomas decomposition [9, 10] for the PDE system.This talk is an extension of the methodology of [8, 11, 12, 13, 14]. As a relevantexample in practice, we apply the procedure of the strong consistent FDA gen-eration to the two-dimensional Navier-Stokes equations for the unsteady motionof an incompressible fluid of constant viscosity. For these equations, we construct

73

On Strongly Consistent Finite Difference Approximations 3

two fully conservative FDAs (one s-consistent and one w-consistent). We use theFDAs for the numerical simulation on exact solutions and consider a Kármán vor-tex street to analyze the influence of the consistency on the numerical quality ofthese schemes.

References

[1] Godunov, S.K., Ryaben’kii, V.S.: Difference schemes. An introduction to the under-lying theory. Elsevier, New York (1987).

[2] Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations.An Introduction, Cambridge University Press, (2005).

[3] Samarskii, A.A.: Theory of Difference Schemes. Marcel Dekker, New York (2001).

[4] Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2ndEdition. SIAM, Philadelphia (2004).

[5] Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods,2nd Edition. Springer, New York (1998).

[6] Gerdt, V.P., Robertz, D.: Computation of Difference Groebner Bases. Comp. Sci. J.of Moldova 20, No.2(59), 203–226 (2012).

[7] Levin, A.: Difference Algebra. Algebra and Applications, Vol. 8. Springer (2008).

[8] Gerdt, V.P.: Consistency Analysis of Finite Difference Approximations to PDE Sys-tems. Mathematical Modelling in Computational Physics / MMCP 2011, Adam,G., Buša, J., Hnatič, M. (Eds.), LNCS 7125, pp. 28–42. Springer, Berlin (2012).arXiv:math.AP/1107.4269

[9] Bächler, T., Gerdt, V., Lange-Hegermann, M. and Robertz, D.: Algorithmic Thomasdecomposition of algebraic and differential systems. Journal of Symbolic Computa-tion, 47(10), 1233–1266 (2012). arXiv:math.AC/1108.0817

[10] Robertz, D.: Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathe-matics 2121. Springer, Cham (2014).

[11] Amodio, P., Blinkov Yu.A., Gerdt, V.P. and La Scala, R.: Algebraic construction andnumerical behavior of a new s-consistent difference scheme for the 2D Navier–Stokesequations. Applied Mathematics and Computation, 314, pp.408-421. (2017).

[12] Gerdt, V.P., Blinkov, Yu.A., Mozzhilkin, V.V.: Groebner Bases and Generationof Difference Schemes for Partial Differential Equations. SIGMA 2, 051 (2006).arXiv:math.RA/0605334

[13] Gerdt, V.P., Blinkov, Yu.A.: Involution and Difference Schemes for the Navier-StokesEquations. Computer Algebra in Scientific Computing / CASC 2009, Gerdt, V.P.,Mayr, E.W., Vorozhtsov, E.V. (Eds.), LNCS 5743, pp. 94–105. Springer, Berlin (2009).

[14] Gerdt, V.P., Robertz, D.: Consistency of Finite Difference Approximations for Lin-ear PDE Systems and its Algorithmic Verification. Watt, S.M. (Ed.), Proceedings ofISSAC 2010, pp. 53–59. Association for Computing Machinery (2010).

[15] Kármán, T.: Aerodynamics. McGraw-Hill, New York (1963).

74

4 Dominik Michels, Vladimir Gerdt, Dmitry Lyakhov and Yuri Blinkov

Dominik MichelsVisual Computing CenterKing Abdullah University of Science and TechnologyThuwal, Saudi Arabiae-mail: [email protected]

Vladimir GerdtLaboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubna, Russiae-mail: [email protected]

Dmitry LyakhovVisual Computing CenterKing Abdullah University of Science and TechnologyThuwal, Saudi Arabiae-mail: [email protected]

Yuri BlinkovFaculty of Mechanics and MathematicsSaratov State UniversitySaratov, Russian Federatione-mail: [email protected]

75

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77

On the chordality of polynomial sets in triangular

decomposition in top-down style

Chenqi Mou

Abstract. In this talk, we show the connections between chordal graphs whichpermit perfect elimination orderings on their vertexes from Graph Theoryand triangular decomposition which decompose polynomial sets into trian-gular sets from Computer Algebra and present the chordal graph structuresof polynomial sets appearing in triangular decomposition in top-down stylewhen the input polynomial set has a chordal associated graph. In particular,we show that the associated graph of one specific triangular set in any algo-rithm for triangular decomposition in top-down style is a subgraph of thatchordal graph and that all the triangular sets computed by Wang’s methodfor triangular decomposition have associated graphs which are subgraphs ofthat chordal graph. Furthermore, the associated graphs of polynomial setscan be used to describe their sparsity with respect to the variables, and wepresent a refined algorithm for efficient triangular decomposition for sparsepolynomial sets in this sense.

This talk is based on the joint work with Yang Bai.

1. Chordal graphs and triangular decomposition

Let K be a field, and K[x] be the multivariate polynomial ring over K in thevariables x1, . . . , xn.

For a polynomial F ∈ K[x], define the (variable) support of F , denoted bysupp(F ), to be the set of variables in x1, . . . , xn which effectively appear in F . Fora polynomial set F ⊂ K[x], supp(F) := ∪F∈F supp(F ), and its associated graph

G(F) = (V,E) is an undirected graph with V = supp(F) and E = (xi, xj) :∃F ∈ F such that xi, xj ∈ supp(F ).

Let G = (V,E) be a graph with V = x1, . . . , xn. Then an ordering xi1 <xi2 < · · · < xin of the vertexes is called a perfect elimination ordering of Gif for each j = i1, . . . , in, the restriction of G on the set xj ∪ xk : xk <xj and (xk, xj) ∈ E is a clique. A graph G is said to be chordal if there ex-ists a perfect elimination ordering of it and a polynomial set F ⊂ K[x] is said tobe chordal if G(F) is chordal.

78

2 Chenqi Mou

For example, the associated graph of P = x2 + x1, x3 + x1, x24 + x2, x

34 +

x3, x5 + x2, x5 + x3 + x2 is shown in Figure 1. One can find that the associatedgraph G(P) is chordal by definition and thus P is chordal.

Figure 1. The associated graphs G(P)

Let the variables in K[x] be ordered as x1 < · · · < xn. An ordered setof non-constant polynomials T ⊂ K[x] is called a triangular set if the greatestvariables of the polynomials in T increase strictly. A finite number of triangularsets T1, . . . , Tr ⊂ K[x] are called a triangular decomposition of a polynomial setF ⊂ K[x] if Z(F) = ∪r

i=1(Z(Ti)\Z(∏

T∈Tiini(T )) holds, where ini(T ) is the leading

coefficient of T with respect to the greatest variable of T and Z(·) denotes the setof common zeros.

Roughly speaking, an algorithm A for computing triangular decompositionof F ⊂ K[x] is said to be in top-down style if the elimination of variables inA follows a strict order xn, xn−1, . . . , x1 and in the process of eliminating eachxi (1 ≤ i ≤ n), no variables greater than xi (namely xi+1, . . . , xn) are generated.

2. Main theoretical results

2.1. General algorithms for triangular decomposition in top-down style

Denote the power set of a set S by 2S . For an integer i (1 ≤ i ≤ n), let fi be amapping

fi : 2K[xi]\K[xi−1] → (K[xi] \K[xi−1])× 2K[xi−1]

P 7→ (T,R)(1)

such that supp(T ) ⊂ supp(P) and supp(R) ⊂ supp(P). For a polynomial setP ⊂ K[x] and a fixed integer i (1 ≤ i ≤ n), suppose that (Ti,Ri) = fi(P

(i)) forsome fi as stated above. For j = 1, . . . , n, define the polynomial set

redi(P(j)) :=

P(j), if j > iTi, if j = i

P(j) ∪R(j)i , if j < i

79

Chordality of Polynomial Sets 3

and redi(P) := ∪nj=1 redi(P

(j)). In particular, write

redi(P) := redi(redi+1(· · · (redn(P)) · · · )) (2)

for simplicity.The mapping fi in (1) is abstraction of specific reductions with respect to

one variable xi used in different kinds of algorithms for triangular decompositionin top-down style.

Theorem 2.1. Let F ⊂ K[x] be a chordal polynomial set with x1 < · · · < xn as one

perfect elimination ordering and redi(F) be defined in (2) for i = n, . . . , 1. Then

the following statements hold:

(a) For each i = n, . . . , 1, G(redi(F)) ⊂ G(F).(b) If T := red1(F) does not contain any nonzero constant, then T forms a

triangular set such that G(T ) ⊂ G(F).

Theorem 2.1 (b) tells us that under the conditions stated in the theorem,the associated graph of one specific triangular set computed in any algorithm fortriangular decomposition in top-down style is a subgraph of the associated graphof the input polynomial set. In fact, this triangular set is the “main branch” in thetriangular decomposition in the sense that other branches are obtained by addingadditional constrains in the splitting in the process of triangular decomposition.

2.2. Wang’s method for triangular decomposition

A simply-structured algorithm was proposed byWang for triangular decompositionin top-down style in 1993 [2]. The decomposition process in Wang’s method appliedto a polynomial set F ⊂ K[x] can be viewed as a binary tree with its root as(F , ∅, n).

Theorem 2.2. Let F ⊂ K[x] be a chordal polynomial set with x1 < · · · < xn as

one perfect elimination ordering and (P,Q, i) be any node in the binary decompo-

sition tree for Wang’s method applied to F . Then G(P) ⊂ G(F). In particular, let

T1, . . . , Tr be the triangular sets computed by Wang’s method. Then G(Ti) ⊂ G(F)for i = 1, . . . , r.

As shown by Theorem 2.2, with a chordal input polynomial set, all the poly-nomials in the nodes of the decomposition tree of Wang’s method, and thus allthe computed triangular sets, have associated graphs which are subgraphs of thatof the input polynomial set.

3. Fast triangular decomposition for variable sparse polynomial sets

Let G(F) = (V,E) be the associated graph of a polynomial set F ⊂ K[x]. Thenthe variable sparsity sv(F) of F can be defined as

sv(F) = |E|/

(

2

|V |

)

,

80

4 Chenqi Mou

where the denominator is the number of edges of a complete graph composed of|V | vertexes. Triangular decomposition of a chordal and variable sparse polynomialset F ⊂ K[x] with an algorithm in top-down style can be sped up by using theperfect elimination ordering of the chordal associated graph G(F).

Some experimental comparisons of timings for computing regular decompo-sition of one class of chordal and variable sparse polynomials [1]

Fi := xkxk+3 − xk+1xk+2 : k = 1, 2, . . . , i , i ∈ Z>0

with respect to the perfect elimination ordering versus random orderings are re-ported in the following table, where n denotes the variable number in Fi, svdenotes the variable sparsity, tp and tr are the timings (in seconds) for regulardecomposition with respect to the perfect elimination orderings and 5 randomlychosen variable orderings respectively, and tr are the average timings for randomorderings.

Table 1. Timings for regular decomposition of Fi

n sv tp tr tr tr/tp8 0.64 0.11 0.10 0.09 0.05 0.06 0.09 0.10 0.9110 0.53 0.19 0.14 0.21 0.22 0.11 0.21 0.18 0.9520 0.28 1.44 4.24 4.45 3.15 4.41 4.65 4.18 2.9025 0.23 4.25 50.62 20.29 15.55 25.01 35.10 29.31 6.9030 0.19 11.94 177.37 185.94 130.54 142.97 103.42 148.05 12.4035 0.17 42.33 560.56 291.35 633.43 320.98 938.45 548.95 12.9740 0.15 161.11 1883.64 3618.04 4289.13 4013.99 2996.37 3360.23 20.86

References

[1] Diego Cifuentes and Pablo A Parrilo. Chordal networks of polynomial ideals. SIAM

J. Appl. Algebra Geom., 1(1):73–110, 2017.

[2] Dongming Wang. An elimination method for polynomial systems. J. Symbolic Com-

put., 16(2):83–114, 1993.

Chenqi MouLMIB – School of Mathematics and Systems ScienceBeijing Advanced Innovation Center for Big Data and Brain ComputingBeihang University, Beijing 100191, Chinae-mail: [email protected]

81

On complexity of trajectories in the equal-mass

free-fall three-body problem

Mylläri Aleksandr, Mylläri Tatiana, Myullyari Anna and Vassiliev

Nikolay

Abstract. We study complexity of trajectories in the equal-mass free-fallthree-body problem. We construct numerically symbolic sequences using dif-ferent methods: close binary approaches, triple approaches, collinear configu-rations and other. Different entropy estimates for individual trajectories andfor a system as a whole are compared.

We analyse complexity of trajectories in the equal-mass free-fall three-bodyproblem by numerically constructing symbolic sequences and calculating differententropy-like parameters for these sequences. We also discuss some ways to estimateKolmogorov and Kolmogorov-Sinai entropy.

Symbolic dynamics was used to analyze some special cases of the three-bodyproblem: Alexeyev [2, 3, 4, 5] has found an intermittence of motions of differ-ent types in the one special case of the three-body problem - Sitnikov problem.Symbolic dynamics was also applied in two other special cases of the three-bodyproblem: the rectilinear problem (Tanikawa & Mikkola [10, 11]); and the isoscelesproblem (Zare & Chesley [14, 6]). Tanikawa & Mikkola [12] considered the casewith non-zero angular momentum; they also studied free-fall case and have foundsequences of triple collision orbits and periodic orbits for isosceles and collinearcases [13].

It is not easy to visualize initial conditions in the general case because ofthe high dimension of the problem: 3 masses of the bodies + 9 initial coordinates+ 9 initial velocities. Equal-mass free-fall three-body problem is much easier andconvenient for study: it drastically reduces the dimension of the problem andallows easy visualization of initial configuration. All the masses are equal, so allpermutations of the bodies will give us equivalent systems. Since initial velocitiesare zero, the problem becomes flat, and at any moment of time we have only twocomponents of coordinates and velocities for each body. If at the initial moment weplace two bodies in the points (−0.5; 0) and (0.5; 0), then all possible configurationswill be covered if we place the third body inside the region D bounded by two

82

2 Mylläri Aleksandr, Mylläri Tatiana, Myullyari Anna and Vassiliev Nikolay

M3

-0.4 -0.2 0.2 0.4

0.2

0.4

0.6

0.8

Figure 1. Agekian-Anosova region D.

straight line segments and arc of the unit circle centered at (−0.5, 0) (Fig. 1) [1].All other possible initial configurations (with zero initial velocities) can be receivedby the projection to this region D and (if needed) transformation of time.

We used code by Seppo Mikkola (Tuorla Observatory, University of Turku)[8] for numerical simulations.

Typical final stage of the evolution of three-body system is close binary mov-ing in one direction, while the third body moves in the opposite direction. So,of interest is finite segment of the symbolic sequences while (infinite) final partsof these sequences are predictable (the only difference is which of the bodies isejected). If one will calculate entropy of such "infinite" sequence, the result is ob-vious. So, we study the evolution of the system during the finite period of time(anyway, we can not integrate it infinitely long), considering the stage of activeinteraction between the bodies. This way, we study complexity of finite sequencesand in our "numerical symbolic dynamics" approach we replace original three-body system by a dynamical system that behaves like our original system duringthis period of time, and have similar behavior all other time (without disruption).

We scan region D with step 0.0005 on each coordinate. For each startingpoint we numerically integrate equations of motion, construct symbolic sequencesand estimate entropy of each sequence. One can use different methods to constructsymbolic sequences (see e.g. [9]). In this study, we construct symbolic sequencesusing binary encounters (we detect minimum distance between two bodies, andcorresponding symbol is the number of the distant body, i.e. symbols are fromthe alphabet 1, 2, 3), triple encounters (we detect minimum of the sum of allthree mutual distances between bodies, corresponding symbol is the number ofthe distant body, again symbols are from the alphabet 1, 2, 3), using collinearconfiguration (similar to [13] we detect the moment when one body crosses theline connecting two distant bodies, there are 6 different configurations possible)and projection to the Agekian-Anosova region D (in [7] authors call it homologymapping – there are 6 possible combinations of projecting our three bodies to theregion D, thus the alphabet in this case is 1, 2, 3, 4, 5, 6).

We also estimate Kolmogorov-Sinai entropy using same approach as in [7] –study spreading of projection of neighboring trajectories on the homology map,

83

Complexity of trajectories in the equal-mass free-fall three-body problem 3

but while in [7] authors used a "drop" consisting of 100 initial points, we use onlynine points (point under consideration and 8 neighbours around it in our grid ofinitial conditions).

References

[1] Agekian, T.A. and Anosova, J.P. 1967, Astron. Zh., 44, 1261

[2] Alexeyev, V. M. 1968a, Math. sbornik, 76, 72

[3] Alexeyev, V. M. 1968, Math. sbornik, 77, 545

[4] Alexeyev, V. M. 1969, Math. sbornik, 78, 3

[5] Alexeyev, V. M. 1981, Uspekhi math. nauk, 36, 161

[6] Chesley, S. 1999, Celest. Mech. Dyn. Astron., 73, 291

[7] Heinämäki, P., Lehto, H., Valtonen. M., and Chernin A., 1999, MNRAS, 310, 811

[8] Mikkola, S. and Tanikawa, K. 1999, Celest. Mech. Dyn. Astron., 74, 287-295.

[9] Mylläri, A., Orlov, V., Chernin, A., Martynova, A. and Mylläri, T. 2016, Baltic As-tronomy, vol. 25, 254

[10] Tanikawa, K. and Mikkola, S. 2000a, Cel. Mech. Dyn. Astron., 76, 23

[11] Tanikawa, K. and Mikkola, S. 2000b, Chaos, 10, 649

[12] Tanikawa, K. and Mikkola, S. 2007, in the Proceedings of the Workshop held in St.Petersberg, August 25 - 30, 2007, eprint arXiv:0802.2465, 02/2008

[13] Tanikawa, K. and Mikkola, S. 2015, Publ. Astr. Soc. Japan, vol. 67, No. 6, 115 (1-10)

[14] Zare, K. and Chesley, S. 1998, Chaos, 8, 475

Mylläri AleksandrDept. of Computers & Technology, SASSt.George’s UniversitySt.George’s, Grenada, West Indiese-mail: [email protected]

Mylläri TatianaDept. of Computers & Technology, SASSt.George’s UniversitySt.George’s, Grenada, West Indiese-mail: [email protected]

Myullyari AnnaAccendo Data LLCCoral Springs,Florida, USAe-mail: [email protected]

Vassiliev NikolayV.A. Steklov Institute of Mathematicsof the Russian Academy of SciencesSt. Petersburg, Russiae-mail: [email protected]

84

Factorization Method for the Second-Order Lin-

ear Nonlocal Difference Equations

I.N. Parasidis and E. Providas

Abstract. First, we present solvability criteria and a formula for construct-

ing closed-form solutions to arbitrary second-order linear difference equations

with variable coefficients and nonlocal multipoint boundary conditions. Next,

we develop an operator factorization method for solving exactly boundary

value problems for second-order linear difference equations with polynomial

coefficients and containing up to the three boundary points. Of particular

relevance here are the references [1, 2, 3].

1. Introduction

Denote by S the linear space of all real-valued functions (sequences) uk = u(k), k ∈

N. Let A : S → S be a second-order linear difference operator defined by

Auk = uk+2 + akuk+1 + bkuk, (1.1)

where ak, bk, uk ∈ S and bk 6= 0 for all k ≥ k1 or preferably for k = 1, . . .. In

addition, let the operator A : S → S be defined as

Auk = Auk,

D(A) = uk ∈ S : µi1u1 + µi2u2 + . . .+ µi,lul = βi, i = 1, 2, l ≥ 2, (1.2)

where µij , βi ∈ R, i = 1, 2, j = 1, . . . , l; that is to say A is a restriction of A

denoted compactly by A ⊂ A.

Let u(1)k , u

(2)k be a fundamental solution set of the homogeneous equation

Auk = 0 and u(fk)k be a particular solution of the non-homogeneous equation

85

2 I.N. Parasidis and E. Providas

Auk = fk, fk ∈ S. Introduce the vector u(H)k = (u

(1)k u

(2)k ) and the associated

Casorati matrix along with the vectors

C0 =

(u(1)1 u

(2)1

u(1)2 u

(2)2

), u0 =

(u1

u2

), u

(fk)0 =

(u(fk)1

u(fk)2

). (1.3)

Furthermore, consider the equation Auk = fk for k = 1, . . . l− 3 together with the

two nonlocal boundary conditions and define the l × l matrix

D =

b1 a1 1 0 · · · 0 0 0

0 b2 a2 1 · · · 0 0 0...

. . .. . .

. . .. . .

. . .. . .

. . .

0 · · · 0 · · · · · · bl−2 al−2 1

µ11 µ12 · · · · · · · · · µ1,l−2 µ1,l−1 µ1,l

µ21 µ22 · · · · · · · · · µ2,l−2 µ2,l−1 µ2,l

, (1.4)

and the vectors

ul =

u1

u2

...

ul−2

ul−1

ul

=

(u0

u2

), u2 =

u3

...

ul

, βf =

f1

f2...

fl−2

β1

β2

. (1.5)

Then the following theorem holds.

Theorem 1.1. If detD 6= 0, then ul = D−1bf and the nonlocal boundary value

problem

Auk = fk (1.6)

admits a unique solution which can be obtained in closed-form as

uk = u(fk)k + u

(H)k C−1

0 (u0 − u(fk)0 ). (1.7)

The application of Theorem 1.1 requires the analytic form of two linearly

independent solutions and a particular solution of the corresponding homogeneous

and non-homogeneous equations, respectively, which may be very difficult to obtain

in many cases with variable coefficients. Alternatively, we can use a factorization

method.

86

Factorization Method for the Second-Order Linear Nonlocal Difference... 3

2. Factorization Method

Definition 2.1. A second-order linear difference operator A defined by (1.1) is said

to be factorable when it can be written as a product (composition) of two first-

order linear operators A1, A2 : S → S, viz.

Auk = A1A2uk. (2.1)

Lemma 2.2. An operator A defined by (1.1) is factorable when there exist rk, sk ∈ S

such that

Auk = yk+1 + rkyk, (2.2)

A1yk = yk+1 + rkyk, A2uk = yk, (2.3)

where yk = uk+1+skuk. Moreover, rk, sk are a solution of the difference equations

sk+1 + rk = ak,

skrk = bk. (2.4)

We confine our investigations to the cases where the coefficients ak, bk are

polynomials and there exist polynomials rk, sk which satisfy the system of equa-

tions (2.4).

Theorem 2.3. Let ak, bk be polynomials of degree Deg ak and Deg bk, respectively.

Then the second-order operator A is factorable in the following cases:

(i) If Deg ak < Deg bk and there exists a polynomial sk of degree Deg sk =

0 or 1 . . . orDeg bk satisfying the equation

sksk+1 − aksk + bk = 0, (2.5)

or

(ii) If Deg ak = Deg bk and there exists a polynomial sk of degree Deg sk = 0 or

Deg sk = Deg bk satisfying Eq. (2.5),

Then the polynomial sk can be constructed by the method of undetermined coeffi-

cients and thus rk = ak − sk+1.

Now we state the main theorem in this paper.

Theorem 2.4. Let the second-order linear difference operator A defined by (1.2)

with l = 3, viz.

Auk = uk+2 + akuk+1 + bkuk,

D(A) = uk ∈ S : µi1u1 + µi2u2 + µi3u3 = βi, i = 1, 2. (2.6)

87

4 I.N. Parasidis and E. Providas

Further, let rk, sk solve the system of difference equations (2.4). If

detD =

b1 a1 1

µ11 µ12 µ13

µ21 µ22 µ23

6= 0, (2.7)

then,

(i) The operator A can be factored to A = A1A2 where the injective first-order

operators A1 and A2 are defined by

A1yk = yk+1 + rkyk = fk, D(A1) = yk ∈ S : y1 = u∗

2 + s1u∗

1, (2.8)

A2uk = uk+1 + skuk = y∗k, D(A2) = uk ∈ S : u1 = u∗

1, (2.9)

where yk = uk+1+skuk, Auk = A1yk, u∗

3 = col(u∗

1, u∗

2, u∗

3), bf = col(f1, β1, β2)

and u∗

3 = D−1bf , and y∗k = A−1

1 fk.

(ii) The unique solution of the three-point boundary value problem is given in

closed-form by

uk = A−1fk = A−12 A−1

1 fk = A−12 y∗k. (2.10)

Finally, we solve the next example problem.

Example 2.5. The operator A : S → S defined by

Auk = uk+2 − (k + 2)uk+1 + (k + 1)uk = (k + 1)! ,

D(A) = uk ∈ S : u1 − u2 + 2u3 = 4, 2u1 + u2 + u3 = 5, (2.11)

is injective and the unique solution of (2.11) is given by the formula

uk =5

4+

k−1∑

j=1

j!(j −

3

2

)(2.12)

3. Conclusion

The technique presented here is simple to use, it can be easily incorporated to any

Computer Algebra System (CAS) and more important it can be extended to deal

with more complicated problems embracing nonlocal boundary conditions with

many points and non-polynomial variable coefficients.

88

Factorization Method for the Second-Order Linear Nonlocal Difference... 5

References

[1] A.F. Voevodin, Factorization method for linear and quasilinear singularly perturbed

boundary value problems for ordinary differential equations, Numer. Analys. Appl.

(2009), Vol. 2, No. 1, pp. 1–12.

[2] A.Dobrogowska, G.Jakimowicz, Factorization method applied to the second order dif-

ference equations, Appl. Math. Lett. (2017), 74, pp. 161–166.

[3] I.N. Parasidis and E. Providas, Closed-form solutions for some classes of loaded dif-

ference equations with initial and nonlocal multipoint conditions, in Modern Discrete

Mathematics and Analysis, T. Rassias and N.J. Daras, eds., Springer, 2018.

I.N. Parasidis

Dept. of Electrical Engineering

TEI of Thessaly

41110 Larissa, Greece


E. Providas

Dept. of Mechanical Engineering

TEI of Thessaly

41110 Larissa, Greece


89

Usage of Automatic Differentiation in Some

Practical Problems of Celestial Mechanics

Dmitry Pavlov

Abstract. Building numerically integrated orbits (ephemeris) of celestial bod-ies has been for a long time an area of celestial mechanics with rich outcomein terms of both science and technology. The model of ephemeris contains alarge number of initial parameters and constants that are determined fromobservations and have an uncertainty. The algorithm requires the first-orderderivatives of orbital parameters w.r.t all the determined parameters in thewhole timespan of observations. One of the approaches of obtaining thosederivatives, examined in this work, is the integration of the derivatives simul-taneously with the equations of motion. That requires calculating a functionand its partial derivatives w.r.t. a number of parameters at the same time,which is essentially the case for the automatic differentiation technique.

Another usage of the automatic differentiation is the propagation ofuncertainty of initial parameters and constants to orbits; the uncertainty,which generally grows with time, can be estimated via the (time-dependent)Jacobian matrix obtained with the numerical integration.

On another note, automatic differentiation allows to build a numericalintegrator that is not based on difference schemes like the traditional methodsused in celestial mechanics (see papers of Jorba and Zou on Taylor method).

Some preliminary practical results are presented.

Dmitry PavlovLaboratory of Ephemeris AstronomyInstitute of Applied Astronomy of the Russian Academy of SciencesSt. Petersburg, Russiae-mail: [email protected]

90

A sharp version of Shimizu’s theorem on entire

automorphic functions

Ronen Peretz

Abstract. This paper develops further the theory of the automorphic groupof non-constant entire functions. This theory essentially started with two re-markable papers of Tatsujiro Shimizu that were published in 1931. There arethree results in this paper. The first result is that the Aut(f)-orbit of anycomplex number has no finite accumulation point. The second result is an ac-curate computation of the derivative of an automorphic function of an entirefunction at any of its fixed points. The third result gives the precise form ofan automorphic function that is uniform over an open subset of C. This lastresult is a follow up of a remarkable theorem of Shimizu. It is a sharp formof his result. It leads to an algorithm of computing the entire automorphicfunctions of entire functions. The complexity is computed using an heightestimate of a rational parameter discovered by Shimizu.

1. Introduction

In 1931 Tatsujiro Shimizu published two remarkable papers having the titles:On the Fundamental Domains and the Groups for Meromorphic Functions. I andII. [2, 3]. There he set up the foundations of the theory of automorphic functions ofmeromorphic functions. If f(w) is a non-constant meromorphic function then theautomorphic functions of f are the solutions φ(z) of the automorphic equation:

f(φ(z)) = f(z). (1.1)

Usually these are many valued functions. They form a group which we denote byAut(f). The binary operation being composition of mappings. Most of the results

2010 Mathematics Subject Classification: 30B40,30C15,30D05,30D20,30D30,30D99,

30F10,30F35,32D05,32D15

Key Words and Phrases: entire functions, integral functions, meromorphic functions, funda-

mental domains, automorphic functions of a meromorphic function, the automorphic group of a

meromorphic function

91

2 Ronen Peretz

of Shimizu in [2, 3], refer to the properties of the individual automorphic functions.In a recent paper, [1], a complementary set of results were obtained. Many of whichrefer to the global structure of the automorphic group, Aut(f), rather than to theproperties of its individual elements. A very interesting result proved by Shimizuasserts that if the automorphic function φ ∈ Aut(f) is uniform over an open subsetof C (no matter how small), then φ(z) must be a linear function of the special formeiθπz + b for some rational θ ∈ Q and some constant b ∈ C. This result is provenin a sequence of theorems: Theorem 11, Theorem 12, Theorem 13 and Theorem14. In fact in Theorem 14 Shimizu proves also the converse, i.e. that for any sucha function φ(z) = eiθπz + b, there exists a meromorphic function f(w), such thatφ ∈ Aut(f). Shimizu uses in his proofs of these theorems some deep results fromthe theory of complex dynamics as developed by Fatou and by Julia as well asthe Iversen method and well known theorems of Gross and Valiron. There is noindication in Shimizu’s theorems as to what are the actual possible values of thearithmetic parameter θ ∈ Q. This gap is closed in the current paper where we getan accurate set of possible values of θ in terms of the orders of the zeros of the firstderivative of f(w). This enables us to compute an upper bound for the height ofShimizu’s parameter θ. An immediate application is an algorithm that computesthe entire automorphic functions of f(w). The complexity of this algorithm caneasily be estimated using our upper bound for the height of θ. That is the thirdresult of our paper. Its proof relies on our second result, which is the computationof the derivative of an automorphic φ at any of its fixed-points. Rather than usingthe machinery of complex dynamics we invoke an elementary approach that usescalculations with power series. This hard-computational approach has the benefitof being constructive and it gives us effective possible values for φ′(z0), for a fixedpoint φ(z0) = z0. That is one of the tools used in our height estimate. Anothertool is Theorem 8.4 in [1] which implies that Z(f ′) = Fix(Aut(f)). The first resultof our paper is really the straight forward observation that the Aut(f)-orbit of anycomplex number can not have a finite accumulation point. This is immediate bythe rigidity property of holomorphic functions. A variant of this was used coupleof times by Shimizu. For convenience, we assume in this paper that f(w) is a non-constant entire function. We denote by E the set of all the non-constant entirefunctions.

2. The main results and their proofs

Proposition 2.1. Let f ∈ E. Then we have:(1) ∀ z ∈ C, the Aut(f)-orbit of z, i.e. the set φ(z) |φ ∈ Aut(f), (where onlythose φ ∈ Aut(f) are taken for which φ(z) is defined) has no finite accumulationpoint.(2) If φ ∈ Aut(f) has a fixed-point z0, then either φ′(z0) = 1 or f ′(z0) = 0.(3) Aut(f) ∩Aut(f ′) ⊂ z + b | b ∈ C.

92

A sharp version of Shimizu’s theorem on entire automorphic functions 3

Proof.

(1) If z ∈ C, φn ∈ Aut(f) are such that the elements of the sequence φn(z)nare different from one another and limn→∞ φn(z) = b ∈ C exists, then: f(z) =f(φ1(z)) = f(φ2(z)) = . . . = f(φn(z)) = . . . = f(b), where the last equality followsby the continuity of f . This implies that f(w) ≡ f(b), a constant. This contradictsthe assumption that f ∈ E and in particular that f is not a constant function.(2) The automorphic equation f(φ(z)) = f(z) implies that φ(z) · f ′(φ(z)) = f ′(z).In the last identity we take the limit z → z0 and recall the assumption φ(z0) = z0.The result obtained is φ′(z0) · f

′(z0) = f ′(z0). If f′(z0) 6= 0 then φ′(z0) = 1.

(3) If φ ∈ Aut(f) ∩ Aut(f ′), then f(φ(z)) = f(z) and φ′(z) · f ′(z) = f ′(z) (byφ′(z) · f ′(φ(z)) = φ′(z) · f ′(z)). Hence φ′(z) ≡ 1.

Theorem 2.2. Let f ∈ E, φ ∈ Aut(f) has a fixed-point z0, and f ′(z0) = . . . =f (n−1)(z0) = 0, while f (n)(z0) 6= 0. Then:

φ′(z0) ∈

e2πik/n | k = 0, . . . , n− 1

.

Proof.

We use the following expansions about z0:

φ(z) = z0 + φ′(z0)(z − z0) + . . . , φ′(z) = φ′(z0) + φ′′(z0)(z − z0) + . . . ,

f ′(z) =f (n)(z0)

(n− 1)!(z − z0)

n−1 + . . . ,

f ′(φ(z)) = f ′(z0 + φ′(z0)(z − z0) + . . .) =f (n)(z0)

(n− 1)!(φ′(z0)(z − z0) + . . .)n−1 + . . . .

We substitute these into the identity φ′(z)f ′(φ(z)) = f ′(z):

(φ′(z0) + φ′′(z0)(z − z0) + . . .)

(

f (n)(z0)

(n− 1)!(φ′(z0)(z − z0) + . . .)n−1 + . . .

)

=

=f (n)(z0)

(n− 1)!(z − z0)

n−1 + . . . .

Equating the coefficients of the lowest non-vanishing power of (z−z0) which turnsup to be (z − z0)

n−1 gives:

φ′(z0)f (n)(z0)

(n− 1)!(φ′(z0))

n−1 =f (n)(z0)

(n− 1)!.

Hence (φ′(z0))n = 1 which proves the assertion.

Remark 2.3. Theorem 2.2 is a more accurate version of Proposition 2.1(2).

We can, now, strengthen Theorem 13 on page 247 of [3]. Here is that result:

Theorem 13. [3] A rational integral function Φ(z) can not satisfy the equationf(Φ(z)) = f(z) for a meromorphic (transcendental) function f(z), unless Φ(z) is

93

4 Ronen Peretz

a linear function of the form eiθπz + b, θ being a rational number.

We also recall that Shimizu demonstrated that if Φ ∈ Aut(f) and if there is anopen subset V ⊆ C over which Φ is uniform, then Φ(z) = eiθπz+ b for some θ ∈ Q

and some b ∈ C. Thus, the family of these linear functions are the only possibleentire functions that qualify as automorphic functions. Here is our sharper versionwhich bounds from above the height of the rational number θ ∈ Q in terms of theorders of the zeros of the derivative f ′(z).

Theorem 2.4. If f ∈ E and if Φ ∈ Aut(f) and Φ is uniform over some non-emptyopen subset ∅ 6= V ⊆ C, then Φ(z) = eiθπz + b for some θ ∈ Q and some b ∈ C

where either θ ≡ 0 mod (2π) or b1−eiθπ

∈ Z(f ′) in which case if:

f ′

(

b

1− eiθπ

)

= . . . = f (n−1)

(

b

1− eiθπ

)

= 0, f (n)

(

b

1− eiθπ

)

6= 0, n ≥ 2,

then:

θ ∈

2k

n| k = 0, 1, . . . , n− 1

.

Proof.

Since Φ(z) is uniform on some non-empty open subset ∅ 6= V ⊆ C, it follows bythe results of Shimizu mentioned above that Φ(z) = eiθπz + b for some θ ∈ Q andsome b ∈ C. If θ 6≡ 0 mod (2π) it follows that eiθπ 6= 1, and that:

Φ

(

b

1− eiθπ

)

=b

1− eiθπ,

a fixed-point of the automorphic function Φ(z). By Theorem 8.4 of [1] we have:Z(f ′) = Fix(Aut(f)). Hence:

f ′

(

b

1− eiθπ

)

= 0.

Clearly, there should exist a smallest n ∈ Z+, n ≥ 2 such that:

f (n)

(

b

1− eiθπ

)

6= 0.

Otherwise f(w) ≡ Const. which contradicts the assumption f ∈ E. By Theorem2.2 above we have:

θ ∈

2k

n| k = 0, 1, . . . , n− 1

.

Theorem 2.4 is now proved.

Remark 2.5. By Theorem 2.4 it follows that height(θ) is at most equals the orderof the zero of the function:

f(z)− f

(

b

1− eiθπ

)

at z =

(

b

1− eiθπ

)

,

94

A sharp version of Shimizu’s theorem on entire automorphic functions 5

minus 1.

Thus the following problem is solvable by an algorithm of complexity that couldeasily be estimated apriori (in the worst case scenario):

Input: An entire function f ∈ E and a zero z0 of its derivative, i.e. f ′(z0) = 0.

Output: Determine if f(z) has an entire automorphic function Φ(z) related toz0. If such an automorphic function exists, then compute it.

The algorithm:

Step 1. Compute the order n of the zero of the function f(z)− f(z0) at z = z0. Itmust satisfy n ≥ 2 by the input.Step 2. Loop on k = 1, . . . , n − 1. For each k compute the complex number

bk = z0(1− e2πik/n). Check if the following functional equation is satisfied:

f(e2πik/nz + bk) = f(z).

If it is satisfied, then output Φ(z) = e2πik/nz + bk. Stop!Step 3. Output: ”No such an automorphic function exists!”.

References

[1] R. Peretz, On the automorphic group of an entire function, submitted for publication,August, 2017.

[2] Tatsujiro Shimizu, On the Fundamental Domains and the Groups for MeromorphicFunctions. I, Japanese journal of mathematics: transactions and abstracts, Volume 8,pp 175-236, 1931.

[3] Tatsujiro Shimizu, On the Fundamental Domains and the Groups for MeromorphicFunctions. II, Japanese journal of mathematics: transactions and abstracts, Volume8, pp 237-304, 1931.

Ronen PeretzDepartment of MathematicsBen Gurion University of the NegevBeer-Sheva , 84105IsraelE-mail: [email protected]

Ronen Peretz

95

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103

Computational Linear and Commutative Algebra

Lorenzo Robbiano

A new book written with Martin Kreuzer will be described in my talk.

• From the back cover of the book:

This book combines, in a novel and general way, an extensive development of

the theory of families of commuting matrices with applications to zero-dimensional

commutative rings, primary decompositions and polynomial system solving. It

integrates the Linear Algebra of the Third Millennium, developed exclusively here,

with classical algorithmic and algebraic techniques. Even the experienced reader

will be pleasantly surprised to discover new and unexpected aspects in a variety

of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces

of commuting families of endomorphisms, multiplication maps of zero-dimensional

affine algebras, computation of primary decompositions and maximal ideals, and

solution of polynomial systems.

This book completes a trilogy initiated by the uncharacteristically witty

books Computational Commutative Algebra 1 and 2 by the same authors. The

material treated here is not available in book form, and much of it is not available

at all. The authors continue to present it in their lively and humorous style, inter-

spersing core content with funny quotations and tongue-in-cheek explanations.

• From the review of David A. Cox:

– This book is a lovely blend of commutative and linear algebra.

– The book contains many new results and concepts, along with known ideas

drawn from a widely scattered literature.

References

[1] M. Kreuzer and L. Robbiano, Computational Linear and Commutative Algebra.Springer, 2016

Lorenzo RobbianoDipartimento di MatematicaUniversità di GenovaGenova, Italye-mail: [email protected]

104

Confluent Heun equation and equivalent first-

order systems

A.A. Salatich and S. Yu. Slavyanov

Introduction

Presented text is an enlargement and elaboration of other publication of theauthors [1]. The new vector formulations of confluent Heun equation (furtherCHE) is proposed. In its turn the text specifies integral symmetries and relationto Painleve equations as obtained in [2].

Consider CHE with two Fuchsian singularities at finite points zj , j = 1, 2and an irregular singularity at infinity. It reads

L1(D, z)w(z) = (σ(z)D2 + τ(z)D + (ω(z)− th))w(z) = 0, (1)

Here either

σ(z) = z(z − 1)

τ(z) = −z(z − 1) + c(z − 1) + dz

ω(z) = −az (2)

or

σ(z) = z(z − t)

τ(z) = −z(z − t) + c(z − t) + dz

ω(z) = −az (3)

In both cases (2), (3) polynomials σ(z) and τ(z) are of second degree in z. Asthe result, differential operator L1 has dimension 1 according to [3]. Note that thechosen form of CHE (1), (2) corresponds to that in the book [4] however (1), (3)is different from it. The advantage of the latter presentaton is discussed in [1].

Parameter h is called the accessory parameter. The chosen factor in front ofit, namely t, leads to to the following lemma.

Lemma 1. Equation (1) with (3) is reduced to confluent hypergeometricequation at t = 0.

105

2 A.A. Salatich and S. Yu. Slavyanov

Proof. Set t = 0 in (1) with (3). We obtain instead of (1) the confluenthypergeometric equation.

At choosing another factor not proportional t, the accessory parameter h isconserved in limiting equation.

The interest to CHE is growing in last decades [4, 5]. Firstly, it is moregeneral comparative to confluent hypergeometric equation. Secondly, more andmore physical applications arise.

Linear first order system

The confluent Heun equation can be linked to first-order linear systems. However,these links can be different. One possible way of choosing such a system isdetermined by the demand that the the residues at Fuchsian points have zerodeterminant. In the other approach traces of these residues are taken to be zero.We study the first case here. Let the first-order system be

~Y ′(z) = A(z) ~Y (z), T (z) = σ(z)A(z) (4)

where

A(z) =A(1)

z+

A(2)

z − t+A(∞) (5)

with

A(1) =

(

0 0h θ1

)

A(2) =

(

a11 a12a21 θ2 − a11

)

A(∞) =

(

t 00 0

)

The condition

detA(2) = 0

implies

a11(Θ2 − a11)− a21a12 = 0

Hence, we arrive to the following result for matrix T

T (z) =

(

a11z + tz(z − t) a12z

h(z − t) + a21z θ1(z − t) + (θ2 − a11)z

)

(6)

Further computations give

trT = tz(z − t) + θ1(z − t) + θ2z

detT = σ(z)(a11θ1 − a12h− t2θ1 + t(θ2 + θ1 − a11)z)

T12

(

T11

T12

)

′

= tz

In view of lemma 1 the matrix element a12 should be chosen as

a12 = t (7)

The searched equation for the first component of vector ~Y (z) reads

σ(z)y′′1 (z) + P (z)y′1(z) +Q(z)y1(z) = 0 (8)

106

Confluent Heun equation and equivalent first-order systems 3

where

P (z) = −σ(z)

(

lnT12

σ

)

′

− trT = −tz(z − t) + (θ1 + 1)z + θ2(z − t)

Q(z) = T12

(

T11

T12

)

′

+ σ(z)−1detT =

zt(θ1 + θ2 − a11 + 1) + a11θ1 − th

The followinng relations between matrix elements and parameters of equation (1)hold

a = a11 − θ1 − θ2, c = θ2, d = θ1 + 1 (9)

Shift in accessory parameter is not essential.

Painleve equation PV

Painleve equation is a nonlinear integrable equation, widely studied and applied inlast decades. Recent researches in this field one can find, for instance, in collectionof papers [8]. Our interests lay in bijection relation between Heun equations andPainleve equations. [4, 7].

The approach presented in this paper serves as justification of heuristicantiquantization of Heun equation proposed in previous papers starting withpublication in J. Phys. A.: Math. Gen. [9].

We shortly repeat derivation of PV . The transformation of a Hamiltonian toa Lagrangian consist of transfer from variable µ to variable q and transfer fromHamiltonian H(µ, q) to Lagrangian L(q, q) according to

q =∂H

∂µ=

2σ(q)µ+ τ(q)

t

L(q, q) = qµ−H(µ, q) =

((t)1/2q − (t)−1/2τ(q))2

4σ(q)−

ω(q)

t(10)

The corresonding Euler equation is actially PV however it is not completely equalto traditional form of PV . In order to find and hence resolve the discrepancy weperform the inverse transformation to variable q → qt. That means returning totraditional form of CHE. We obtain

q −1

2

(

1

q+

1

q − 1

)

q2 +q

t−

1

2t2

[

(c2 + 1)q

q − 1− d2

q − 1

q

]

−

1

2q(q − 1)(2q − 1) +

q(q − 1)

t((c+ d)− 2a) = 0 (11)

The derived equation is a subset of Painleve equation PV . The discussion ofits generality can be found in [4].

107

4 A.A. Salatich and S. Yu. Slavyanov

References

[1] S. Yu. Slavyanov, A. A. Salatich Confluent Heun equation and confluent

hypergeometric equation, (in Russian ) Zap. Nauchn. Sem. PDMI 462 93-102, (2017).

[2] A.Ya. Kazakov, S. Yu. Slavyanov, TMPh. 179, 543-549, (2014).

[3] S. Slavyanov, TMPh., 193, 401-408, (2017).

[4] S. Slavyanov, W. Lay, Special functions: A unified theory based on singularities,Oxford University Press, Oxford - New-York, 2000

[5] Ed. A. Ronveau, Heun’s differential equation (Oxford New York Tokyo, OxfordUniversity Press), 1995.

[6] S. Slavyanov, D. Shatko, A. Ishkhanyan, T. Rotinyan, TMPh., 189, 1726-1733 (2016).

[7] S. Yu. Slavyanov, O. L. Stesik, TMPh, 186, 118-125, (2016).

[8] A. D. Bruno, A. B. Batkin (eds.), Painleve Equations and Related Topics, DeGruyter, 2012.

[9] S. Yu. Slavyanov, J. Phys. A, 29, 7329-7335, (1996).

A.A. Salatich and S. Yu. SlavyanovSaint Petersburg State University,Saint Petersburg, Russiae-mail: [email protected]

108

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110

On the Pierce–Birkhoff conjecture and relatedproblems.

Mark Spivakovsky

Let R be a real closed field and B = R[x1, . . . , xn] a polynomial ring over Rin n variables.

Definition 0.1. A function g : Rn → R is said to be piecewise polynomial if Rn

can be covered by a finite collection of closed semi-algebraic sets Pi, i ∈ 1, . . . , ssuch that for each i there exists a polynomial gi ∈ B satisfying g|Pi

= gi|Pi.

Piecewise polynomial functions form a ring, containing B, which is denotedby PW (B).

Consider the ring (contained in PW (B)) of all the functions obtained fromB by iterating the operations of sup and inf. The Pierce–Birkhoff conjecture wasstated by M. Henriksen and J. Isbell in the early nineteen sixties ([1] and [3]):

Conjecture 1. (Pierce-Birkhoff) If g : Rn → R is in PW (B), then there exists afinite family of polynomials gij ∈ B such that f = sup

i

infj(gij) (in other words, for

all x ∈ Rn, f(x) = supi

infj(gij(x))).

In this talk, we will recall the definition of the real spectrum of a ring Σ,denoted by Sper Σ. In the nineteen eighties, generalizing the problem from thepolynomial ring to an arbitrary ring Σ, J. Madden proved that the Pierce–Birkhoffconjecture for Σ is equivalent to a statement about an arbitrary pair of pointsα, β ∈ Sper Σ and their separating ideal < α, β >; we refer to this statementas the local Pierce-Birkhoff conjecture at α, β. In [4] we introduced a strongerconjecture, also stated for a pair of points α, β ∈ Sper Σ and the separatingideal < α, β >, called the Connectedness conjecture, about a finite family ofelements f1, . . . , fr ∈ Σ. In [6] we introduced a new conjecture, called the StrongConnectedness conjecture, and proved that the Strong Connectedness conjecturein dimension n− 1 implies the strong connectedness conjecture in dimension n inthe case when ht(< α, β >) ≤ n− 1.

The Pierce-Birkhoff Conjecture for r = 2 is equivalent to the ConnectednessConjecture for r = 1; this conjecture is called the Separation Conjecture. The

111

2 Mark Spivakovsky

Strong Connectedness Conjecture for r = 1 is called the Strong Separation Con-jecture. In this talk fix a polynomial f ∈ R[x, z] where x = (x1, . . . , xn), z are n+1independent variables. We will define the notion of two points α, β ∈ Sper R[x, z]being in good position with respect to f . Our main result is a proof of the StrongSeparation Conjecture in the case when α and β are in good position with respectto f . We also prove that, given a connected semi-algebraic set D ⊂ Rn, if thenumber of real roots of f , counted with or without multiplicity, is constant forall x ∈ D then these roots are represented by continuous semi-algebraic functionsφj : D → R.

References

[1] G. Birkhoff and R. Pierce, Lattice-ordered rings. Annales Acad. Brasil Cienc. 28,41–69 (1956).

[2] M. Henriksen and J. Isbell, On the continuity of the real roots of an algebraic equa-

tion, Proc. AMS Vol. 4, pp. 431-434, 1953.

[3] M. Henriksen and J. Isbell, Lattice-ordered rings and function rings. Pacific J. Math.11, 533–566 (1962).

[4] F. Lucas, J.J. Madden, D. Schaub and M. Spivakovsky, On connectedness of sets in

the real spectra of polynomial rings, Manuscripta Math. 128, 505-547, 2009.

[5] F. Lucas, D. Schaub and M. Spivakovsky, Approximate roots of a valuation and the

Pierce-Birkhoff Conjecture, Ann. Fac. Sci. Toulouse, Mathematique, Serie 6, Vol.XXI, Fasc. 2, 259-342, 2012.

[6] F. Lucas, D. Schaub and M. Spivakovsky, On the Pierce-Birkhoff Conjecture, Journalof Algebra 435, (2015), 124-158.

[7] J. J. Madden, Pierce–Birkhoff rings. Arch. Math. 53, 565–570 (1989).

Mark SpivakovskyCNRS, Institut de Mathematiques de Toulouse 118,route de Narbonne, F-31062 Toulouse Cedex 9, Francee-mail: [email protected]

112

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ComplexInfinity ,

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118

Reverse Decomposition of Unipotents

Nikolai Vavilov

Abstract. Decomposition of unipotents gives short polynomial expressions ofthe conjugates of elementary genetators as products of elementaries. It turnsout that with some minor twist the decomposition of unipotents can be readbackwards, to give very short polynomial expressions of elementary generatorsin terms of elementary conjugates of an arbitrary matrix and its inverse. Forabsolute elementary subgroups of classical groups this was recently observedby Raimund Preusser. I discuss various generalisations of these results forexceptional groups, at the relative level, and possible applications.

Decomposition of unipotents [6] was first proposed by Alexei Stepanov forGL(n,R) in 1987, immediately generalised to other split classical groups by thepresent author, and then further developed in other contexts by a number ofauthors, see [8, 5, 2] for many further references.

In its simplest form, it can be viewed as a constructive version of the normalityof the elementary subgroup. Namely, let Φ be a root system, R be an arbitrarycommutative ring with 1, and G(Φ, R) be the simply connected Chevalley groupof type Φ over R. Further, fix a split maximal torus T (Φ, R) of G(Φ, R) and thecorresponding elementary generators xα(ξ), where α ∈ Φ, ξ ∈ R. Let E(Φ, R) bethe elementary subgroup spanned by all these elementary generators.

Then decomposition of unipotents provides explicit polynomial formulae ex-pressing the conjugate gxα(ξ)g

−1 of an elementary generator by an arbitrary ma-trix g ∈ G(Φ, R) as a product of elementaries. Thus, for instance, for the groupsof types E6 and E7 any such conjugate is the product of at most 4 · 27 · 16 and4 · 56 · 27 elementary generators, respectively [7]

Another central classical result in the structure theory of Chevalley groupsis description of their normal subgroups, or rather their subgroups normalised bythe elementary group E(Φ, R). What would be an explicit contructive version ofthat? Until very recently, this was only known in some very special cases. Thus,for SL(n,Z), n ≥ 3, Joel Brenner [1] established that for an arbitrary matrixg ∈ SL(n,Z) an elementary transvection tij(ξ), where ξ belongs to the level of g,is a bounded product of conjugates of g and g−1. Brenner’s proof used the theoryof elementary divisors, and even generalisations to other groups over PID were

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2 Nikolai Vavilov

not immediate at all. And of course, there was no hope whatsoever to write suchsimilar formulae for arbitrary commutative rings.

Thus, we were seriously perplexed, when we’ve first seen the preprints of [3, 4]in Summer 2017. The calculations in [3] start in exactly the same way as in [6],so predictably our assessment of these papers came through the following threestages: 1) There must be nothing new as compared with [6], 2) Gosh, why is it trueat all? 3) It is a fantastic breakthrough in the structure theory of algebraic-likegroups!

Technically, the twist introduced by Raimund Preusser in the decomposi-tion of unipotents seems to be minor. It consists in expressing a conjugate of anelementary generator not as a product of factors sitting in proper parabolics ofcertain types, but rather sitting in the products of these parabolics by somethingsmall in the unipotent radicals of the opposite parabolics. We were aware of theidea itself [5], but have never appreciated the whole significance of this apparentlysmall variation.

In fact, it allows to reduce degree of the resulting polynomials, and thusboth to completely avoid the cumbersome “main lemma”, establishing that thecoefficients of the occuring polynomials generate the unit ideal, and drasticallylower the depth of commutators. In particular, Preusser’s idea allows to proveanalogues of Brenner’s lemma for groups of all types over arbitrary commutativerings, and more.

Immediately after understanding this idea, we were able to generalise it toexceptional groups as well, and to other situations. In particular, it can be derivedthat for an arbitrary commutative ring R and an arbitrary matrix g ∈ G(Φ, R)one can write explicit formulae, expressing an elementary generator xα(ξ), whereξ belongs to the level of g, as products of at most 8 ·dim(G) elementary conjugatesof g and g−1.

I discuss further development of this idea, such as our joint paper with ZhangZuhong, where we write similar formulae at the relative level, expressing elemen-tary generators as products of conjugates of g and g−1 by elements of the relativeelementary subgroups E(Φ, R, I), corresponding to an ideal I ER. This result hassignificant applications to the description of subnormal subgroups of G(Φ, R), etc.

I sketch further imminent applications of these ideas, to description of variousclasses of intermediate subgroups, the values of word maps, etc.

References

[1] J. L. Brenner, The linear homogeneous group. III, Ann. of Math. (2) 71 (1960), 210—223.

[2] V.A. Petrov, Decomposition of transvections: An algebro-geometric approach,, Algebrai Analiz 28 (2016), no. 1, 150–157.

[3] R. Preusser, Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited, J.Group Theory 21 (2018), 21–44

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[4] R. Preusser, Sandwich classification for O2n+1(R) and U2n+1(R,∆) revisited,arXiv:1801.00699v1 [math.KT] 2 Jan 2018, 1–20, accepted to J. Group Theory (2018)

[5] A.V.Stepanov, A new look at the decomposition of unipotents and the normal structure

of Chevalley groups, Algebra i Analiz 28 (2016), no. 3, 161–173.

[6] A. Stepanov and N. Vavilov, Decomposition of transvections: A theme with variations,K-Theory 19 (2000), no. 2, 109–153.

[7] N. Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova, 104(2000), 201–250.

[8] N.A. Vavilov, Decomposition of unipotents for E6 and E7: 25 years after, Zap. Nauchn.Sem. POMI 430 (2014), 32–52.

Nikolai VavilovDepartment of mathematics and MechanicsSt. Petersburg State UniversitySt. Petersburg, Russiae-mail: [email protected]

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