Editorial Board P. Flajolet, M. Ismail, E. Lutwak Volume...

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

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Editorial Board

P. Flajolet, M. Ismail, E. Lutwak

Volume 99

Solving Polynomial Equation Systems II

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Cambridge University Press0521811562 - Solving Polynomial Equation Systems II: Macaulay’s Paradigm and GrobnerTechnologyTeo MoraFrontmatterMore information

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FOUNDING EDITOR G.-C. ROTAEditorial BoardP. Flajolet, M. Ismail, E. Lutwak

40 N. White (ed.) Matroid Applications41 S. Sakai Operator Algebras in Dynamical Systems42 W. Hodges Basic Model Theory43 H. Stahl and V. Totik General Orthogonal Polynomials45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions46 A Bjorner et al. Oriented Matroids47 G. Edgar and L. Sucheston Stopping Times and Directed Processes48 C. Sims Computation with Finitely Presented Groups49 T. Palmer Banach Algebras and the General Theory of *-Algebras I50 F. Borceux Handbook of Categorical Algebra I51 F. Borceux Handbook of Categorical Algebra II52 F. Borceux Handbook of Categorical Algebra III53 V. F. Kolchin Random Graphs54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics57 P. M. Cohn Skew Fields58 R. Gardner Geometric Tomography59 G. A. Baker, Jr., and P. Graves-Morris Pade Approximants, 2nd edn60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity Theory61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory63 A. C. Thompson Minkowski Geometry64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications65 K. Engel Sperner Theory66 D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of Graphs67 F. Bergeron, G. Labelle and P. Leroux Combinatorial Species and Tree-Like Structures68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry71 G. E. Andrews, R. Askey and R. Roy Special Functions72 R. Ticciati Quantum Field Theory for Mathematicians73 M. Stern Semimodular Lattices74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II76 A. A. Ivanov Geometry of Sporadic Groups I77 A. Schinzel Polymomials with Special Regard to Reducibility78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2nd edn79 T. Palmer Banach Algebras and the General Theory of *-Algebras II80 O. Stormark Lie’s Structural Approach to PDE Systems81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets83 C. Foias et al. Navier–Stokes Equations and Turbulence84 B. Polster and G. Steinke Geometries on Surfaces85 R. B. Paris and D. Kaminski Asymptotics and Mellin–Barnes Integrals86 R. McEliece The Theory of Information and Coding, 2nd edn87 B. Magurn Algebraic Introduction to K-Theory88 T. Mora Solving Polynomial Equation Systems I89 K. Bichteler Stochastic Integration with Jumps90 M. Lothaire Algebraic Combinatorics on Words91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II92 P. McMullen and E. Schulte Abstract Regular Polytopes93 G. Gierz et al. Continuous Lattices and Domains94 S. Finch Mathematical Constants95 Y. Jabri The Mountain Pass Theorem







Solving Polynomial Equation Systems II

Macaulay’s Paradigm and Grobner Technology

TEO MORA

University of Genoa






C A M B R I D G E U N I V E R S I T Y P R E S S

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In the beginning was the Word, and the Word was with God, and the Word was God.St John (Authorized Version)

God bless the girl who refuses to study algebra. It is a study that has caused many a girlto lose her soul.Superintendent Francis of the Los Angeles schools.

The present state of our knowledge of the properties of Modular Systems is chiefly dueto the fundamental theorems and processes of L. Kronecker, M. Noether, D. Hilbert, andE. Lasker, and above all to J. Konig’s profound exposition and numerous extensions ofKronecker’s theory. Konig’s treatise might be regarded as in some measure complete ifit were admitted that a problem is finished with when its solution has been reduced toa finite number of feasible operations. If however the operations are too numerous ortoo involved to be carried out in practice the solution is only a theoretical one; and itsimportance then lies not in itself, but in the theorems with which it is associated and towhich it leads. Such a theoretical solution must be regarded as a preliminary and notthe final stage in the consideration of the problem.F. S. Macaulay, The Algebraic Theory of Modular Systems

Gauss is the perfect representative of the Thaurus mathematicians. Their style consistsin performing long and numerous computations until this allows them to guess a con-jecture, usually a correct one.Theodyl Magus, Astrology and Mathematics






Contents

Preface page xiSetting xiv

Part three: Gauss, Euclid, Buchberger: ElementaryGrobner Bases 1

20 Hilbert 320.1 Affine Algebraic Varieties and Ideals 320.2 Linear Change of Coordinates 820.3 Hilbert’s Nullstellensatz 1020.4 *Kronecker Solver 1520.5 Projective Varieties and Homogeneous Ideals 2220.6 *Syzygies and Hilbert Function 2820.7 *More on the Hilbert Function 3420.8 Hilbert’s and Gordan’s Basissatze 36

21 Gauss II 4621.1 Some Heretical Notation 4721.2 Gaussian Reduction 5121.3 Gaussian Reduction and Euclidean Algorithm Revisited 63

22 Buchberger 7222.1 From Gauss to Grobner 7522.2 Grobner Basis 7822.3 Toward Buchberger’s Algorithm 8322.4 Buchberger’s Algorithm (1) 9622.5 Buchberger’s Criteria 9822.6 Buchberger’s Algorithm (2) 104

23 Macaulay I 10923.1 Homogenization and Affinization 11023.2 H-bases 114

vi






Contents vii

23.3 Macaulay’s Lemma 11923.4 Resolution and Hilbert Function for Monomial

Ideals 12223.5 Hilbert Function Computation: the

‘Divide-and-Conquer’ Algorithms 13623.6 H-bases and Grobner Bases for Modules 13823.7 Lifting Theorem 14223.8 Computing Resolutions 14623.9 Macaulay’s Nullstellensatz Bound 15223.10 *Bounds for the Degree in the Nullstellensatz 156

24 Grobner I 17024.1 Rewriting Rules 17324.2 Grobner Bases and Rewriting Rules 18324.3 Grobner Bases for Modules 18824.4 Grobner Bases in Graded Rings 19524.5 Standard Bases and the Lifting Theorem 19824.6 Hironaka’s Standard Bases and Valuations 20324.7 *Standard Bases and Quotients Rings 21824.8 *Characterization of Standard Bases in

Valuation Rings 22324.9 Term Ordering: Classification and

Representation 23424.10 *Grobner Bases and the State Polytope 247

25 Gebauer and Traverso 25525.1 Gebauer–Moller and Useless Pairs 25525.2 Buchberger’s Algorithm (3) 26425.3 Traverso’s Choice 27125.4 Gebauer–Moller’s Staggered Linear Bases and

Faugere’s F5 27426 Spear 289

26.1 Zacharias Rings 29126.2 Lexicographical Term Ordering and Elimination Ideals 30026.3 Ideal Theoretical Operation 30426.4 *Multivariate Chinese Remainder Algorithm 31326.5 Tag-Variable Technique and Its Application to

Subalgebras 31626.6 Caboara–Traverso Module Representation 32126.7 *Caboara Algorithm for Homogeneous

Minimal Resolutions 329






viii Contents

Part four: Duality 33327 Noether 335

27.1 Noetherian Rings 33727.2 Prime, Primary, Radical, Maximal Ideals 34027.3 Lasker–Noether Decomposition: Existence 34527.4 Lasker–Noether Decomposition: Uniqueness 35027.5 Contraction and Extension 35627.6 Decomposition of Homogeneous Ideals 36427.7 *The Closure of an Ideal at the Origin 36827.8 Generic System of Coordinates 37127.9 Ideals in Noether Position 37427.10 *Chains of Prime Ideals 37827.11 Dimension 38027.12 Zero-dimensional Ideals and Multiplicity 38427.13 Unmixed Ideals 390

28 Moller I 39328.1 Duality 39328.2 Moller Algorithm 401

29 Lazard 41429.1 The FGLM Problem 41529.2 The FGLM Algorithm 41829.3 Border Bases and Grobner Representation 42629.4 Improving Moller’s Algorithm 43229.5 Hilbert Driven and Grobner Walk 44029.6 *The Structure of the Canonical Module 444

30 Macaulay II 45130.1 The Linear Structure of an Ideal 45230.2 Inverse System 45630.3 Representing and Computing the Linear

Structure of an Ideal 46130.4 Noetherian Equations 46630.5 Dialytic Arrays of M(r) and Perfect Ideals 47830.6 Multiplicity of Primary Ideals 49230.7 The Structure of Primary Ideals at the Origin 494

31 Grobner II 50031.1 Noetherian Equations 50131.2 Stability 50231.3 Grobner Duality 50431.4 Leibniz Formula 50831.5 Differential Inverse Functions at the Origin 50931.6 Taylor Formula and Grobner Duality 512






Contents ix

32 Grobner III 51732.1 Macaulay Bases 51832.2 Macaulay Basis and Grobner Representation 52132.3 Macaulay Basis and Decomposition of Primary Ideals 52232.4 Horner Representation of Macaulay Bases 52732.5 Polynomial Evaluation at Macaulay Bases 53132.6 Continuations 53332.7 Computing a Macaulay Basis 542

33 Moller II 54933.1 Macaulay’s Trick 55033.2 The Cerlienco–Mureddu Correspondence 55433.3 Lazard Structural Theorem 56033.4 Some Factorization Results 56233.5 Some Examples 56933.6 An Algorithmic Proof 574

Part five: Beyond Dimension Zero 58334 Grobner IV 585

34.1 Nulldimensionalen Basissatze 58634.2 Primitive Elements and Allgemeine Basissatz 59334.3 Higher-Dimensional Primbasissatz 59834.4 Ideals in Allgemeine Positions 60134.5 Solving 60534.6 Gianni–Kalkbrener Theorem 608

35 Gianni–Trager–Zacharias 61435.1 Decomposition Algorithms 61535.2 Zero-dimensional Decomposition Algorithms 61635.3 The GTZ Scheme 62235.4 Higher-dimensional Decomposition Algorithms 63135.5 Decomposition Algorithms for Allgemeine Ideals 634

35.5.1 Zero-dimensional Allgemeine Ideals 63435.5.2 Higher-dimensional Allgemeine Ideals 637

35.6 Sparse Change of Coordinates 64035.6.1 Gianni’s Local Change of Coordinates 64135.6.2 Giusti–Heintz Coordinates 645

35.7 Linear Algebra and Change of Coordinates 65035.8 Direct Methods for Radical Computation 65435.9 Caboara–Conti–Traverso Decomposition

Algorithm 65835.10 Squarefree Decomposition of a

Zero-dimensional Ideal 660






x Contents

36 Macaulay III 66536.1 Hilbert Function and Complete Intersections 66636.2 The Coefficients of the Hilbert Function 67036.3 Perfectness 678

37 Galligo 68637.1 Galligo Theorem (1): Existence of Generic Escalier 68637.2 Borel Relation 69737.3 *Galligo Theorem (2): the Generic Initial Ideal

is Borel Invariant 70637.4 *Galligo Theorem (3): the Structure of the

Generic Escalier 71037.5 Eliahou–Kervaire Resolution 714

38 Giusti 72538.1 The Complexity of an Ideal 72638.2 Toward Giusti’s Bound 72838.3 Giusti’s Bound 73338.4 Mayr and Meyer’s Example 73538.5 Optimality of Revlex 741

Bibliography 749Index 758






Preface

If you HOPE that this second SPES volume preserves the style of the previ-ous volume, you will not be disappointed: in fact it maintains a self-containedapproach using only undergraduate mathematics in this introduction to ele-mentary commutative ideal theory and to its computational aspects,1 while myhorror vacui compelled me to report nearly all the relevant results in computa-tional algebraic geometry that I know about.

When the commutative algebra community was exposed, in 1979, to Buch-berger’s theory and algorithm (dated 1965) of Grobner bases2, the more alertresearchers, mainly Schreyer and Bayer, immediately realized that this injec-tion of Grobner technology was all one needed to make effective Macaulay’sparadigm for reducing computational problems for ideals either to the cor-responding combinatorial problem for monomials3 or to a more elementarylinear algebraic computation.4 This realization gave to researchers a straight-forward approach which led them, within more or less fifteen years, to com-pletely effectivize commutative ideal theory.

This second volume of SPES is an eyewitness report on this successful in-troduction of effective methods to algebraic geometry.

Part three, Gauss, Euclid, Buchberger: Elementary Grobner Bases, introducesat the same time Buchberger’s theory of Grobner bases, his algorithm for com-puting them and Macaulay’s paradigm.

While I will discuss in depth both of the classical main approaches to theintroduction of Grobner bases – their relation with rewriting rules and the

1 Up to the point that some results whose proof requires knowldge in advanced commutativealgebra are simply quoted, pointing only to the original proof.

2 And to the independent discovery by Spear.3 The computation of the Hilbert function by means of Macaulay’s Lemma (Corollary 23.4.3).4 Macaulay’s notion of H-basis (Definition 23.2.1) and his related lifting theorem (Theo-

rem 23.7.1) transformed by Schreyer as the tool for computing resolution.

xi






xii Preface

Knuth–Bendix Algorithm, and their connection with Macaulay’s H-bases andHironaka’s standard bases as tools for lifting properities to a polynomial al-gebra from its graded algebra – my presentation stresses the relation of boththe notion and the algorithm to elementary linear algebra and Gaussian re-duction; an added bonus of this approach is the ability to link Buchberger’salgorithm with the most recent alternative linear algebra approach proposedby Faugere.

The discussion of Buchberger’s algorithm aims to present what essentiallyis its ‘standard’ structure as can be found in most good implementations.

In the same mood, the discussion of Macaulay’s paradigm is illustrated byshowing how Grobner bases can be applied in order to successfully computethe Hilbert function and the minimal resolution of a finitely generated polyno-mial ideal and to present the most effective algorithmic solutions.

This part also includes Spear’s tag-variable technique, its application in ef-fectively performing ideal operations (intersection, quotient, colon, saturation),Sweedler’s application of them to the study of subalgebras, Erdos’s character-ization of term orderings, the Bayer–Morrison analysis of the state polytopeand the Grobner fan of an ideal.

The next chapter, Noether, is the keystone of the book: it introduces the termi-nology and preliminary results needed to discuss multivariate ‘solving’: theLasker–Noether decomposition theory, extension/contraction of decomposi-tion, the notions of dimension and multiplicity, the Kredel–Weispfenning al-gorithm for computing dimension.

Part four, Duality, discusses linear algebra tools for describing and computingthe multiplicity of both m-primary and m-closed ideals, m being the max-imal at the origin; this includes Moller’s algorithm, its application to solvethe FGLM-problem, the Cerlienco–Mureddu algorithm, and the linear alge-bra structure of configurations of points; but the main section of this part isa careful presentation of Macaulay’s results on inverse systems and a recentalgorithm which computes the inverse system of any m-primary ideal given byany basis.

Part five, Beyond Dimension Zero, begins with a discussion of Grobner’sBasissatze which describe the structure of lexicographical Grobner bases ofprime, primary and radical ideals and their ultimate generalization, Gianni–Kalkbrener’s Theorem; this allows us to specify what it means to ‘solve’ amulti-dimensional ideal and introduces the decomposition algorithms.






Preface xiii

This part also discusses Macaulay’s results on Hilbert functions and perfect-ness, Galligo’s theorem, and Giusti’s analysis of the complexity of Grobnerbases.As congedo I chose the most elegant result within computational commutativealgebra, the Bayer and Stillman proof of the optimality of degrevlex orderings.

It being my firm belief that the best way of understanding a theory and analgorithm is to verify it through a computation, as in the previous volume,the crucial points of the most relevant algorithms are illustrated by examples,all developed via paper-and-pencil computations; readers are encouraged tofollow them and, better, to test their own examples.

In order to help readers to plan their journey through this book, some sectionscontaining only some interesting digressions are indicated by asterisks in thetable of contents.

A possible short cut which allows readers to appreciate the discussion, with-out becoming too bored by the details, is Chapters 20–23, 26–28, 34–35.

I wish to thank Miguel Angel Borges Tranard, Maria Pia Cavaliere, FrancescaCioffi and Franz Pauel for their help, but I feel strongly indebted to MariaGrazia Marinari for her steady support. Also I need to thank all the friends withwhom I have shared this exciting adventure of algorithmizing commutativealgebra.






Setting

1. Let k be an infinite, perfect field, where, if p := char(k) �= 0, it is possibleto extract pth roots,1 and let k be the algebraic closure of k. Let us fix an integervalue n and consider the polynomial ring

P := k[X1, . . . , Xn]

and its k-basis

T := {Xa11 · · · Xan

n : (a1, . . . , an) ∈ Nn}.2. We also fix an integer value r ≤ n and consider

the ring K := k(Xr+1, . . . , Xn),the polynomial ring Q := K [X1, . . . , Xr ] andits k-basis W := {Xa1

1 · · · Xarr : (a1, . . . , ar ) ∈ Nr }.

All the notation introduced will also be applied in this setting, substitutingeverywhere n, k,P, T with, respectively, r, K ,Q,W .

3. For each d ∈ N we will set

Td := {t ∈ T : deg(t) = d} and T (d) := {t ∈ T : deg(t) ≤ d}.4. Where we need to use the set of the terms generated by some subsets ofvariables, we denote for each i, j, 1 ≤ i < j ≤ n, T [i, j] the monomialsgenerated by Xi , . . . , X j ,

T [i, j] ={

Xaii · · · X

a jj : (ai , . . . , a j ) ∈ N j−i+1

},

1 This is the general setting considered in this the volume, except for Chapters 37 and 38 wheremoreover char(k) = 0.

These restrictions can be relaxed in most of the volume, but, knowing my absentmindedness,I consider it safer to leave to the reader the responsibility of doing so.

xiv






Setting xv

and T [i, j]d (respectively T [i, j](d)) denotes those terms whose degree isequal to (respectively bounded by) d.

5. Each polynomial f ∈ k[X1, . . . , Xn] is therefore a unique linear combina-tion

f =∑t∈T

c( f, t)t

of the terms t ∈ T with coefficients c( f, t) in k and can be uniquely decom-posed, by setting

fδ :=∑t∈Tδ

c( f, t)t, for each δ ∈ N,

as f = ∑dδ=0 fδ where each fδ is homogeneous, deg( fδ) = δ and fd �= 0 so

that d = deg( f ).

6. Since, for each i, 1 ≤ i ≤ n,

P = k[X1, . . . , Xi−1, Xi+1, . . . , Xn][Xi ],

each polynomial f ∈ P can be uniquely expressed as

f =D∑

j=0

h j (X1, . . . , Xi−1, Xi+1, . . . , Xn)X ji , hD �= 0,

and

degXi( f ) := degi ( f ) := D

denotes its degree in the variable Xi .

In particular (i = n)

f =D∑

j=0

h j (X1, . . . , Xn−1)X jn , hD �= 0, D = degn( f );

the leading polynomial of f is Lp( f ) := hd , and its trailing polynomial isTp( f ) := h0.

7. The support {t ∈ T : c( f, t) �= 0} of f being finite, once a term ordering <

on T is fixed, f has a unique representation as an ordered linear combinationof terms:

f =s∑

i=1

c( f, ti )ti : c( f, ti ) ∈ k \ 0, ti ∈ T , t1 > · · · > ts .

The maximal term of f is T( f ) := t1, its leading coefficient is lc( f ) :=c( f, t1) and its maximal monomial is M( f ) := c( f, t1)t1.






xvi Setting

8. For any set F ⊂ P we denote

• T<{F} := {T( f ) : f ∈ F};• T<(F) := {τT( f ) : τ ∈ T , f ∈ F};• N<(F) := T \ T<(F);• k[N<(F)] := Spank(N<(F))

and we will usually omit the dependence on < if there is no ambiguity.

9. Each series f ∈ k[[X1, . . . , Xn]] is a unique (infinite) linear combination

f =∑t∈T

c( f, t)t

of the terms t ∈ T with coefficients c( f, t) in k; for any subset N ⊂ T we willalso write the subring

k[[N]] :={∑

t∈N

c( f, t)t

}⊂ k[[X1, . . . , Xn]].

10. For each f, g ∈ P such that lc( f ) = 1 = lc(g), we denote

S(g, f ) := lcm(T( f ), T(g)

T( f )f − lcm(T( f ), T(g)

T(g)g.

For any enumerated set {g1, . . . , gs} ⊂ P , such that lc(gi ) = 1 for each i ,we write T(i) := T(gi ) and, for each i, j, 1 ≤ i < j ≤ s

T(i, j) := lcm (T(i), T( j)) ,

S(i, j) := S(gi , g j ) := T(i, j)

T( j)g j − T(i, j)

T(i)gi .

11. For any field k the (n-dimensional) affine space over k, kn , is the set

kn := {(a1, . . . , an), ai ∈ k};and we will denote by 0 ∈ kn the point 0 := (0, . . . , 0) and m :=(X1, . . . , Xn) the maximal ideal at 0.

12. We associate

• to any set F ⊂ P , the algebraic affine variety Z(F) consisting of eachcommon root of all polynomials in F :

Z(F) := {a ∈ kn : f (a) = 0, for all f ∈ F} ⊂ kn;• and to any set Z ⊂ kn , the ideal I(Z) of all the polynomials vanishing in Z:

I(Z) := { f ∈ P : f (a) = 0, for all a ∈ Z} ⊂ P.






Setting xvii

13. For any finite set F := { f1, . . . , fs} ⊂ P the ideal generated by F isdenoted by (F) or ( f1, . . . , fs) and is the set

(F) := ( f1, . . . , fs) :={

s∑i=1

hi fi : hi ∈ P}

.

14. For an ideal f ⊂ P ,

f :=r⋂

i=1

qi

denotes an irredundant primary representation; for each i , pi := √qi is the

associated prime and δ(i) := dim(qi ) is the dimension of the primary qi .

15. For any field k and any n ∈ N we will denote by C(n, k) the n-tuples ofnon-zero elements in k:

C(n, k) := {(c1, . . . , cn) ∈ kn, ci �= 0, for each i}.For each c := (c1, . . . , cν) ∈ C(ν, k), we denote by

Lc : k[X1, . . . , Xν] → k[X1, . . . , Xν]

the map defined by

Lc(Xi ) :={

Xi + ci Xν if i < ν,

cν Xν if i = ν.

16. A term ordering 2 of the semigroup T is called degree compatible if foreach t1, t2 ∈ T

deg(t1) < deg(t2) �⇒ t1 < t2.

The semigroup T will be usually well-ordered by means of

• the lexicographical ordering induced by X1 < X2 < · · · < Xn , which isdefined by:

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j < b j and ai = bi for i > j;• the degrevlex ordering induced by X1 < X2 < · · · < Xn , which is the

degree-compatible term ordering under which any two terms having thesame degree are compared according to

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for i < j.

2 That is a well-ordering and a semigroup ordering.






xviii Setting

17. Let < be a term ordering on T , and I ⊂ P an ideal, and A := P/I.Then, since A ∼= k[N<(I)], for each f ∈ P, there is a unique

g := Can( f, I, <) =∑

t∈N<(I)

γ ( f, t, <)t

such that

g ∈ k[N(I)] and f − g ∈ I.

18. More generally, if I ⊂ P is an ideal, and q = {q1, . . . , qs} is a linearlyindependent set such that P/I = Spank(q), then, for each f ∈ P, there is aunique vector

Rep( f, q) := (γ ( f, q1, q), . . . , γ ( f, qs, q)) ∈ ks

which satisfies

f −∑

j

γ ( f, q j , q)q j ∈ I.

In particular, if N<(I) = {τ1, . . . , τs}, we have, for each f ∈ P,

γ ( f, t, N<(I)) = γ ( f, t, <), for each t ∈ N<(I),

Rep( f, N<(I)) := (γ ( f, τ1, <), . . . , γ ( f, τs, <)) ∈ ks .

19. In the same setting,

M(q) :={ (

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

}

denotes the set of the square matrices defined by the equalities

Xhql =∑

j

a(h)l j q j , for each l, j, h, 1 ≤ l, j ≤ s, 1 ≤ h ≤ n,

in P/I = Spank(q).

20. In general, when we need to discuss homogenization of polynomials, wewill use the notation hP := k[X0, X1, . . . , Xn] and

hT :={

Xa00 Xa1

1 · · · Xann : (a0, a1, . . . , an) ∈ Nn+1

}.

The homogenization/affinization maps are denoted

h− : P → hP and a− : hP → P






Setting xix

and defined by

h f (X1, . . . , Xn) := Xdeg( f )

0 f

(X1

X0, . . . ,

Xn

X0

),

a f (X0, X1, . . . , Xn) := f (1, X1, . . . , Xn).

For any term ordering < on T the homogenization of < is the term-ordering<h on hT defined by

t1 <h t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2) and at1 < at2.

21. For an ideal I ⊂ P we will denote H(T ; I) its Hilbert function; HI(T ) itsHilbert polynomial, which we will represent as

HI(T ) = k0(I)(

T + d

d

)+ k1(I)

(T + d − 1

d − 1

)+ · · · + kd−1(I)(T + 1) + kd(I);

and H(I, T ) its Hilbert series.

22. For a free-module Pm , we usually denote {e1, . . . , em} its canonical basisand

T (m) = {tei , t ∈ T , 1 ≤ i ≤ m}= {Xa1

1 · · · Xann ei , (a1, . . . , an) ∈ Nn, 1 ≤ i ≤ m}

its monomial k-basis.

23. The free-module Pm is transformed into an N-graded module by as-signing, for each i , a degree deg(ei ) := di and considering each element(g1, . . . , gm) ∈ Pm to be homogeneous of degree R if and only if each gi

will be either 0 or a homogeneous polynomial of degree R − di .Therefore each element f ∈ Pm can be uniquely decomposed as f =∑di=1 fi where each fi ∈ Pm is homogeneous of degree i and d = deg( f )

In a similar way, Pm is also transformed into a T -graded module by

• assigning a term ordering < on T and a term ωi ∈ T to each ei ,

• defining

T-deg : T (m) → T by T-deg(tei ) = tωi ,

• and T-deg : P(m) → T as

T-deg( f ) := max<

{T-deg(τ ) : c( f, τ ) �= 0}

for each f = ∑τ∈T(m) c( f, τ )τ ∈ P(m),






xx Setting

• considering T -homogeneous of T -degree ω any element (γ1, . . . , γm) ∈Pm such that for each i

γi ∈ T , and γiωi = ω unless γi = 0.

Each element f ∈ Pm can therefore be uniquely decomposed as f =∑t∈T ft where each ft ∈ Pm is T -homogeneous of T -degree t .If we fix a well-ordering ≺ on T (m) which is compatible with a term-

ordering < on T that is satisfying

t1 ≤ t2, τ1 � τ2 �⇒ t1τ1 � t2τ2,

for each t1, t2 ∈ T , τ1, τ2 ∈ T (m) then for each f = ∑τ∈T(m) c( f, τ )τ ∈ P(m),

its maximal term is the term T( f ) := max≺{τ : c( f, τ ) �= 0}; its leadingcoefficient is lc( f ) := c( f, T( f )) and its maximal monomial is M( f ) :=lc( f )T( f ).

24. Usually a free resolution of a P-module M will be denoted

0 → Prρδρ−→ Prρ−1

δρ−1−→ · · ·Pri+1δi+1−→ Pri

δi−→ Pri−1 · · ·Pr1δ1−→ Pr0

δ0−→ M

(0.1)

25. We will denote

• by GL(n, k) the general linear group, that is the set of all invertible n × nsquare matrices with entries in k,

• by B(n, k) ⊂ GL(n, k) the Borel group of the upper triangular matricesM := (

ci j), that is those such that i > j �⇒ ci j = 0,

• by N (n, k) ⊂ B(n, k) the subgroup of the upper triangular unipotent matri-ces M := (

ci j), that is those such that

i > j �⇒ ci j = 0, and i = j �⇒ ci j = 1.

We will use the shorthand k[Xi j ] and k(Xi j ) to denote, respectively, thepolynomial ring generated over k by the variables

{Xi j , 1 ≤ i ≤ n, 1 ≤ j ≤ n}and its rational function field.

Any matrix

M := (ci j

) ∈ GL(n, k)

describes the linear transformation

M : k[X1, . . . , Xn] → k[X1, . . . , Xn]






Setting xxi

defined by

M(Xi ) =∑

jci j X j for each i.

If we also write for each i ,

Yi := M(Xi ) =∑

jci j X j ,

we obtain a system of coordinates {Y1, . . . , Yn} and a corresponding change ofcoordinates

k[Y1, . . . , Yn] = k[X1, . . . , Xn]

which is defined by

f (X1, . . . , Xn) = f(∑

id1i Yi , . . . ,

∑i

dni Yi

)∈ k[Y1, . . . , Yn],

where (di j

) = M−1 ∈ GL(n, k),

denotes the inverse of M .

26. The module P∗ := Homk(P, k) denotes the k-vector space of all k-linearfunctionals � : P → k.

Each k-linear functional � : P → k can be encoded by means of the series∑t∈T

�(t)t ∈ k[[X1, . . . , Xn]]

in such a way that to each such series∑

t∈T γ (t)t ∈ k[[X1, . . . , Xn]] is as-sociated the k-linear functional � ∈ P∗ defined, on each polynomial f =∑

t∈T c( f, t)t , by

�( f ) :=∑t∈T

c( f, t)γ (t).

Module P∗ has a natural structure as P-module, which is obtained by defin-ing, for each � ∈ P∗ and f ∈ P , (� · f ) ∈ P∗ as

(� · f )(g) := �( f g), for each g ∈ P.

27. For each k-vector subspace L ⊂ P∗, let

P(L) := {g ∈ P : �(g) = 0, ∀� ∈ L}and for each k-vector subspace P ⊂ P , let

L(P) := {� ∈ P∗ : �(g) = 0, ∀g ∈ P}.






xxii Setting

28. For each τ ∈ W , M(τ ) : Q → K denotes the morphism defined by

M(τ ) = c( f, τ ) for each f =∑t∈W

c( f, t)t ∈ Q

and set

M := {M(τ ) : τ ∈ W} ⊂ Q∗,

and

∇ρ := SpanK (M(τ )(·) : τ ∈ W(ρ)) ,

for each ρ ∈ N.For each K -vector subspace Λ ⊂ SpanK (M), let

I(Λ) := P(Λ) = { f ∈ Q : �( f ) = 0, for each � ∈ Λ}and, for each K -vector subspace P ⊂ Q, let

M(P) := L(P)∩ SpanK (M) = {� ∈ SpanK (M) : �( f ) = 0, for each f ∈ P}.






Editorial Board P. Flajolet, M. Ismail, E. Lutwak Volume...

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