F. Calogero: Solvable dynam. systems and generations …. - [2] O. Bihun and F. Calogero, ......

F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco /20.10.2016 / page 1/26

Zeros of polynomials, solvable dynamical

systems and generations of polynomials Francesco Calogero

Dipartimento di Fisica, Universita’ di Roma “La Sapienza”

Istituto Nazionale di Fisica Nucleare, Sezione di Roma

Abstract

Recent findings concerning the relations between the coefficients and the

zeros of time-dependent monic polynomials will be reported. They underline a

differential algorithm to compute all the zeros of a generic polynomial and

allow the identification of novel classes of dynamical systems solvable by

algebraic operations, including hierarchies of such Newtonian (“accelerations

equal forces”) problems describing an arbitrary number of nonlinearly

interacting point-particles moving in the complex plane. And the related notion

of generations of (monic) polynomials will be introduced and discussed. Part of

this work has been done with Oksana Bihun and with Mario Bruschi.


References

- [1] F. Calogero, “New solvable variants of the goldfish many-body problem”, Studies Appl. Math. 137 (1), 123-139 (2016); DOI:

10.1111/sapm.12096.

- [2] O. Bihun and F. Calogero, “A new solvable many-body problem of goldfish type”, J. Nonlinear Math. Phys. 23, 28-46 (2016).

- [3] O. Bihun and F. Calogero, “Novel solvable many-body problems”, J. Nonlinear Math. Phys. 23, 190-212 (2016).

- [4] O. Bihun and F. Calogero, “Generations of monic polynomials such that the coefficients of the polynomials of the next

generation coincide with the zeros of polynomial of the current generation, and new solvable many-body problems”, Lett. Math.

Phys. 106 (7), 1011-1031 (2016).

- [5] F. Calogero, “A solvable N-body problem of goldfish type featuring N2 arbitrary coupling constants”, J. Nonlinear Math. Phys.

23, 300-305 (2016).

- [6] F. Calogero, “Three new classes of solvable N-body problems of goldfish type with many arbitrary coupling constants”,

Symmetry 8, 53 (2016).

- [7] M. Bruschi and F. Calogero, “A convenient expression of the time-derivative z n (k)(t), of arbitrary order k of the zero zn(t) of a

time-dependent polynomial pN(z;t) of arbitrary degree N in z, and solvable dynamical systems”, J. Nonlinear Math. Phys. 23, 474-

485 (2016).

- [8] F. Calogero, “Novel isochronous N-body problems featuring N arbitrary rational coupling constants”, J. Math. Phys. 57, 072901

(2016); http://dx.doi.org/10.1063/1.4954851 .

- [9] F. Calogero, “Yet another class of new solvable N-body problems of goldfish type”, Qualit. Theory Dyn. Syst. (in press).

- [10] F. Calogero, “New solvable dynamical systems”, J. Nonlinear Math. Phys. 23, 486-493 (2016).

- [11] F. Calogero, “Isochronous N-body problem of goldfish type featuring 2N arbitrary rational parameters”, Nonlinearity

(submitted to, 2016.09.27).

- [12] F. Calogero, “Integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions”, J. Nonlinear Math. Phys.

(in press as Letter).

- [13] F. Calogero, “New C-integrable and S-integrable systems of nonlinear partial differential equation”, Studies Appl. Math.

(submitted to, 2016.07.26).

- [14] F. Calogero, “Nonlinear differential algorithm to compute all the zeros of a generic polynomial”, J. Math. Phys. 57, 083508 (4

pages) (2016); http://dx.doi.org/10.1063/1.4960821). arXiv:1607.05081v1 [math.CA].

- [15] F. Calogero, “Comment on “Nonlinear differential algorithm to compute all the zeros of a generic polynomial” [J. Math.

Phys. 57, 083508 (2016)]”, J. Math. Phys. (4 pages, in press).

- [16] O. Bihun and F. Calogero, “Generations of solvable discrete-time dynamical systems”, J. Math. Phys. (submitted to,

2016.06.23).

http://dx.doi.org/10.1063/1.4954851

http://dx.doi.org/10.1063/1.4960821

http://arxiv.org/abs/1607.05081v1


Nonlinear differential algorithm to compute all

the zeros of a generic polynomial [14]

𝑃𝑁(𝑧; 𝑐; �̃�) = 𝑧𝑁 + ∑ (𝑐𝑚 𝑧𝑁−𝑚) = ∏(𝑧 − 𝑥𝑛)

𝑁

𝑛=1

𝑁

𝑚=1

Now introduce the t-dependent polynomial

𝑝𝑁(𝑧; �⃗�(𝑡); �̃�(𝑡)) = 𝑧𝑁 + ∑ [𝛾𝑚 (𝑡) 𝑧𝑁−𝑚]

𝑁

𝑚=1

= ∏[𝑧 − 𝑦𝑛(𝑡)]

𝑁

𝑛=1

There holds then the following


Proposition. Consider the following system of N nonlinear first-order differential equations satisfied by the N zeros 𝑦𝑛(𝑡) of this polynomial:

�̇�𝑛(𝑡) = − { ∏ [𝑦𝑛(𝑡) − 𝑦𝑠(𝑡)]−1

𝑁

𝑠=1,𝑠≠𝑛

} ∙

∙ ∑ {�̇�𝑚(𝑡) [𝑐𝑚 − 𝛾𝑚(0)] [𝑦𝑛(𝑡)]𝑁−𝑚}𝑁𝑚=1 ,

𝑓𝑚(𝑇) − 𝑓𝑚(0) = 1, 𝛾𝑚(0) = (−1)𝑚 𝜎𝑚(𝑦 ̃(0)) , 𝑚 = 1, ⋯ , 𝑁 ;

𝜎𝑚(�̃�) = ∑ (𝑦𝑠1 𝑦𝑠2

⋯ 𝑦𝑠𝑚) 1≤𝑠1<𝑠2<⋯<𝑠𝑚≤𝑁 .

Then: 𝑥𝑛 = 𝑦𝑛(𝑇) .


It is thus seen that the zeros 𝑥𝑛 of the polynomial

𝑃𝑁(𝑧; 𝑐; �̃�) can be computed---once the N

coefficients 𝑐𝑚 of this polynomial have been

assigned---via the following procedure. Step one:

choose (arbitrarily!) N complex numbers 𝑦𝑛(0). Step

two: compute the N quantities 𝛾𝑚(0). Step three:

integrate (numerically) the above system of

differential equations from t=0 to t=T, starting from

the N initial data 𝑦𝑛(0), getting thereby the N values

𝑦𝑛(𝑇), which give the sought result, 𝑥𝑛 = 𝑦𝑛(𝑇).

[Subcase: 𝑓𝑚(𝑡) = 𝑡 , �̇�𝑚

(𝑡) = 1, 𝑇 = 1 . ]


An integrable Hamiltonian N-body problem in

the plane featuring N arbitrary functions [16]

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)

𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

− { ∏ [𝜉𝑛(𝑡)−𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∑ {𝑔𝑚

(𝛾𝑚 (𝑡)) [𝜉𝑛(𝑡)]

𝑁−𝑚} ,

𝑁

𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] ,

𝜎𝑚(�̃�) = ∑ (𝜉𝑠1𝜉𝑠2

∙∙∙ 𝜉𝑠𝑚)

1≤𝑠1<𝑠2<⋯<𝑠𝑚≤𝑁

.


Coefficients and zeros of monic polynomials

𝑝𝑁(𝑧; �⃗�; �̃�) = 𝑧𝑁 + ∑ (𝑦𝑚 𝑧𝑁−𝑚)

𝑁

𝑚=1

𝑝𝑁(𝑧; �⃗�; �̃�) = ∏(𝑧 − 𝑥𝑛)

𝑁

𝑛=1

�⃗� is an N-vector: its N components 𝑦𝑚 are the N coefficients of the polynomial

𝑝𝑁(𝑧; �⃗�; �̃�) of degree N in the independent (complex) variable z.

�̃� is an unordered set of N numbers 𝑥𝑛, which are the N zeros of the polynomial

𝑝𝑁(𝑧; �⃗�; �̃�). We denote as �⃗�[𝜇] the N-vector, the N components 𝑥[𝜇],𝑛 of which are

given by the specific permutation--- labeled by the integer index 𝜇 in the range

1 ≤ 𝜇 ≤ 𝑁!---of the N numbers 𝑥𝑛 . Hence, to any given �̃� , there generally

correspond N! different N-vectors �⃗�[𝜇] . We will come back to this notion below.


Identities satisfied by the N coefficients and the

N zeros of a polynomial

𝑦𝑚 = (−1)𝑚 𝜎𝑚(�̃�) , 𝑚 = 1, ⋯ , 𝑁 ;

𝜎𝑚(�̃�) = ∑ (𝑥𝑠1 𝑥𝑠2

⋯ 𝑥𝑠𝑚)

1≤𝑠1<𝑠2<⋯<𝑠𝑚≤𝑁

𝜎1(�̃�) = 𝑥1 + 𝑥2 + ⋯ +𝑥𝑁 ;

𝑁 = 2 ∶ 𝜎1(�̃�) = 𝑥1 + 𝑥2 ;

𝜎2(�̃�) = 𝑥1 𝑥2 = 𝑥2 𝑥1 .


An obvious identity (for monic polynomials)

∑ [𝑦𝑚 (𝑥𝑛)𝑁−𝑚] = −

𝑁

𝑚=1

(𝑥𝑛)𝑁 , 𝑛 = 1, ⋯ , 𝑁 .

Other useful (general) identities

𝜎𝑛,𝑚(�⃗�) = 𝛿𝑛1

+ ∑ (𝑥𝑠1 𝑥𝑠2

⋯ 𝑥𝑠𝑛−1) ;

1≤𝑠1<𝑠2<⋯<𝑠𝑛−1≤𝑁,𝑠𝑗≠𝑚, 𝑗=1,⋯,𝑛−1

∑ [(−1)𝑠 (𝑥𝑛)𝑁−𝑠 𝜎𝑠,𝑚(�⃗�)] = −𝛿𝑛𝑚 ∏ (𝑥𝑛 − 𝑥𝑗) ,𝑁𝑗=1,𝑗≠𝑛

𝑁𝑠=1

𝑛, 𝑚 = 1, ⋯ , 𝑁 .


An important remark

𝜎𝑛,𝑚(�̃�) = 𝛿𝑛1

+ ∑ (𝑥𝑠1 𝑥𝑠2

⋯ 𝑥𝑠𝑛−1)

1≤𝑠1<𝑠2<⋯<𝑠𝑛−1≤𝑁𝑠𝑗≠𝑚, 𝑗=1,⋯,𝑛−1

does not have a clear meaning (because it depends on which is 𝑥𝑚). But the identity

∑[(−1)𝑠 (𝑥𝑛)𝑁−𝑠 𝜎𝑠,𝑚(�̃�)] = −𝛿𝑛𝑚 ∏ (𝑥𝑛 − 𝑥𝑗) ,

𝑁

𝑗=1,𝑗≠𝑛

𝑁

𝑠=1

𝑛, 𝑚 = 1, ⋯ , 𝑁 does make good sense: this identity holds true for any assignment of the N components of the unordered set �̃� .


Example: N=2

�̃� = (𝑎, 𝑏) or �̃� = (𝑏, 𝑎)

If �̃� = (𝑎, 𝑏) then

𝜎1,1(�̃�) = 𝜎1,2(�̃�) = 1 , 𝜎2,1(�̃�) = 𝑏 , 𝜎2,2(�̃�) = 𝑎 ,

The identity: 𝑛 = 𝑚 = 1, −𝑎 + 𝑏 = −(𝑎 − 𝑏)

𝑛 = 1, 𝑚 = 2, −𝑎 + 𝑎 = 0

𝑛 = 2, 𝑚 = 1, −𝑏 + 𝑏 = 0

𝑛 = 𝑚 = 2, −𝑏 + 𝑎 = −(𝑏 − 𝑎) .

If �̃� = (𝑏, 𝑎) : exchange a and b, clearly identity still

true.


Time-dependent monic polynomials

𝜓𝑁 (𝜁; �⃗�(𝑡); 𝜉(𝑡)) = 𝜁𝑁 + ∑ [𝛾𝑚 (𝑡) 𝜁𝑁−𝑚]

𝑁

𝑚=1

𝜓𝑁 (𝜁; �⃗�(𝑡); 𝜉(𝑡)) = ∏[𝜁 − 𝜉𝑛(𝑡)]

𝑁

𝑛=1

�⃗�(𝑡) is a time-dependent N-vector: its N components 𝛾𝑚 (𝑡) are

the N coefficients of the time-dependent polynomial

𝜓𝑁 (𝜁; �⃗�(𝑡); 𝜉(𝑡)) of degree N in the independent (generally

complex) variable 𝜁. 𝜉(𝑡) is an unordered set of N numbers 𝜉𝑛(𝑡),

which are the N zeros of the polynomial 𝜓𝑁 (𝜁; �⃗�(𝑡); 𝜉(𝑡)) (but

they may get ordered via their dependence on time if this

dependence is continuous).


Two very useful formulas [1]

𝜉�̇�(𝑡) = − { ∏ [𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∑ {�̇�𝑚(𝑡) [𝜉𝑛(𝑡)]𝑁−𝑚} ,

𝑁

𝑚=1

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

− { ∏ [𝜉𝑛(𝑡)−𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∑ {�̈�𝑚(𝑡) [𝜉𝑛

(𝑡)]𝑁−𝑚

} ,

𝑁

𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] .

The second-derivative formula is particularly useful and is referred below as (***).


Extension to higher derivatives

These formulas have been extended to the third

and fourth derivatives in [3], to derivatives of

arbitrary order in [7] and to the case of discrete time

in [12]. These results are not reported nor discussed

in this talk, but they are instrumental to demonstrate

the solvable character of some of the dynamical

systems reported below.


Many new solvable models---including isochronous

ones---can be manufactured by taking advantage of

these formulas. For instance from (***):

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

− { ∏ [𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∑ {�̈�𝑚(𝑡) [𝜉𝑛(𝑡)]𝑁−𝑚

} ,𝑁

𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] ,

�̇�𝑚 (𝑡) = (−1)𝑚 ∑ {𝜎𝑛,𝑚[𝜉(𝑡)] �̇�𝑛(𝑡)}𝑁𝑛=1 .

Now assume that a “solvable” dynamical system of

Newtonian type (“acceleration equal force”) reads


�̈⃗�(𝑡) = 𝑓[�⃗�(𝑡); �̇⃗�(𝑡); 𝑡]

or equivalently

�̈�𝑚(𝑡) = 𝑓𝑚[𝛾(𝑡); �̇�(𝑡); 𝑡] , 𝑚 = 1, … , 𝑁 .

Then insert these equations of motion in the right-

hand side of (***). In this manner you obtain a new

“solvable” dynamical system of Newtonian type.

Several new examples are exhibited in references

[1-13], see above. Here we display only a well-

known example and two of those new examples.


The goldfish model

�̈�𝑚(𝑡) = 𝒊 𝜔 �̇�𝑚 (𝑡) ,

𝛾𝑚 (𝑡) = 𝛾𝑚 (0) +𝒊 �̇�𝑚(0) [1−exp(𝒊𝜔𝑡)]

𝜔 ,

𝜉�̈�(𝑡) = 𝒊 𝜔 𝜉�̇�(𝑡) + ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

.

Solution: the N coordinates 𝜉𝑛(𝑡) are the N roots of

∑ [𝜉�̇�(0)

𝜉 − 𝜉𝑛(0)] =

𝒊 𝜔

exp(𝒊 𝜔 𝑡) − 1

𝑁

𝑛=1

.

Isochronous with period 𝑇 = 2𝜋/𝜔---or a, generally

small (much less than N!), integer multiple of T.


The “goldfish-CM” model [2]

�̈�𝑚(𝑡) = −𝜔2 𝛾𝑚 (𝑡) + 2 𝑔2 ∑ [𝛾𝑚 (𝑡) − 𝛾𝑗 (𝑡)]−3

𝑁

𝑗=1,𝑗≠𝑚

;

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

− { ∏ [𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∑ {�̈�𝑚(𝑡) [𝜉𝑛(𝑡)]𝑁−𝑚

} ,𝑁

𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] .

Isochronous with period 𝑇 = 2𝜋/𝜔---or a, generally

small (much less than N!), integer multiple of T.


The “goldfish-CM” model equations of motion [2]

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

− { ∏ [𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∙

∙ ∑ {{ −𝜔2 𝛾𝑚 (𝑡) + 2 𝑔2 ∑ [𝛾𝑚 (𝑡) − 𝛾𝑗 (𝑡)]−3

𝑁

𝑗=1,𝑗≠𝑚

} [𝜉𝑛(𝑡)]𝑁−𝑚} ,

𝑁

𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] ,

𝜎𝑚(�̃�) = ∑ (𝜉𝑠1𝜉𝑠2

∙∙∙ 𝜉𝑠𝑚)

1≤𝑠1<𝑠2<⋯<𝑠𝑚≤𝑁

.


A solvable N-body model featuring N2 arbitrary

coupling constants [5]

�̈�𝑚(𝑡) = − ∑[𝐴𝑚𝑗 𝛾𝑗 (𝑡)]

𝑁

𝑗=1

;

𝜉�̈�(𝑡) = ∑ [2 𝜉�̇�(𝑡) 𝜉�̇�(𝑡)


𝑁

𝑗=1,𝑗≠𝑛

+ { ∏ [𝜉𝑛(𝑡) − 𝜉𝑗(𝑡)]

𝑁

𝑗=1,𝑗≠𝑛

}

−1

∙

∙ ∑ {∑ [𝐴𝑚𝑗 𝛾𝑗 (𝑡)]𝑁

𝑗=1 [𝜉𝑛(𝑡)]𝑁−𝑚} ;𝑁𝑚=1

𝛾𝑚 (𝑡) = (−1)𝑚 𝜎𝑚[𝜉(𝑡)] ,

𝜎𝑚(𝜉) = ∑ (𝜉𝑠1𝜉𝑠2

∙∙∙ 𝜉𝑠𝑚)

1≤𝑠1<𝑠2<⋯<𝑠𝑚≤𝑁

.


These examples are obtained directly from the formula (***). But the process can be iterated over and over again.

This motivated the idea to introduce and investigate the generations of monic polynomials obtained by replacing the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation. [4]


The seed polynomial

𝑝𝑁(0)

(𝑧; �⃗�(0); �̃�(0)) = 𝑧𝑁 + ∑ [𝑦𝑚(0)

𝑧𝑁−𝑚]

𝑁

𝑚=1

,

𝑝𝑁(0)

(𝑧; �⃗�(0); �̃�(0)) = ∏ [𝑧 − 𝑥𝑛(0)

]

𝑁

𝑛=1

.

The first generation of N! monic polynomials

𝑝𝑁(𝜇1;1)

(𝑧; �⃗�[𝜇1](1)

; �̃�[𝜇1](1)

) = 𝑧𝑁 + ∑ [𝑦[𝜇1],𝑚(1)

𝑧𝑁−𝑚] ,

𝑁

𝑚=1

𝑝𝑁(𝜇1;1)

(𝑧; �⃗�[𝜇1](1)

; �̃�[𝜇1](1)

) = ∏ [𝑧 − 𝑥[𝜇1],𝑛(1)

] ,

𝑁

𝑛=1

�⃗�[𝜇1](1)

= �⃗�[𝜇1](0)

; 𝑦[𝜇1],𝑚(1)

= 𝑥[𝜇1],𝑚(0)

.

The integer index 𝝁𝟏 taking values in the range 𝟏 ≤ 𝝁𝟏 ≤ 𝑵! labels the permutations of the a

priori unordered set �̃�(𝟎) .


The second generation of (N!)2 monic

polynomials

𝑝𝑁

(�⃗⃗⃗�(2);2)(𝑧; �⃗�

[�⃗⃗⃗�(2)]

(2) ; �̃�

[�⃗⃗⃗�(2)]

(2)) = 𝑧𝑁 + ∑ [𝑦

[�⃗⃗⃗�(2)],𝑚

(2) 𝑧𝑁−𝑚] ,

𝑁

𝑚=1

𝑝𝑁

(�⃗⃗⃗�(2);2)(𝑧; �⃗�

[�⃗⃗⃗�(2)]

(2) ; �̃�

[�⃗⃗⃗�(2)]

(2)) = ∏ [𝑧 − 𝑥

[�⃗⃗⃗�(2)],𝑛

(2)]

𝑁

𝑛=1

,

�⃗�[�⃗⃗⃗�(2)]

(2)= �⃗�

[�⃗⃗⃗�(2)]

(1)= �⃗�[𝜇1],[𝜇2]

(1); 𝑦

[�⃗⃗⃗�(2)],𝑚

(2)= 𝑥

[�⃗⃗⃗�(2)],𝑚

(1) .

The 2-vector �⃗⃗⃗�(𝟐) = (𝝁𝟏, 𝝁𝟐) has integer components 𝝁𝟏, 𝝁𝟐 taking

values in the range 𝟏 ≤ 𝝁𝟏, 𝝁𝟐 ≤ 𝑵! . The label 𝝁𝟐 identifies the

permutation of the a priori unordered set �̃�[𝝁𝟏](𝟏)

the components of

which are the N zeros of the polynomial of the previous generation

with index 𝝁𝟏, i. e. the N zeros of 𝒑𝑵(𝝁𝟏;𝟏)

(𝒛; �⃗⃗⃗�[𝝁𝟏](𝟏)

; �̃�[𝝁𝟏](𝟏)

) .


The k-th generation of (N!)k monic

polynomials

𝑝𝑁

(�⃗⃗⃗�(𝑘);𝑘)(𝑧; �⃗�

[�⃗⃗⃗�(𝑘)]

(𝑘) ; �̃�

[�⃗⃗⃗�(𝑘)]

(𝑘)) = 𝑧𝑁 + ∑ [𝑦

[�⃗⃗⃗�(𝑘)],𝑚

(𝑘) 𝑧𝑁−𝑚] ,

𝑁

𝑚=1

𝑝𝑁

(�⃗⃗⃗�(𝑘);𝑘)(𝑧; �⃗�

[�⃗⃗⃗�(𝑘)]

(𝑘) ; �̃�

[�⃗⃗⃗�(𝑘)]

(𝑘)) = ∏ [𝑧 − 𝑥

[�⃗⃗⃗�(𝑘)],𝑛

(𝑘)]𝑁

𝑛=1 ,

�⃗�[�⃗⃗⃗�(𝑘)]

(𝑘)= �⃗�

[�⃗⃗⃗�(𝑘)]

(𝑘−1)= �̃�

[�⃗⃗⃗�(𝑘−1)],[𝜇𝑘]

(𝑘−1); 𝑦

[�⃗⃗⃗�(𝑘)],𝑚

(𝑘)= 𝑥

[�⃗⃗⃗�(𝑘)],𝑚

(𝑘−1) .

The k-vector �⃗⃗⃗�(𝒌) = (𝝁𝟏, 𝝁𝟐, … , 𝝁𝒌) has integer components 𝝁𝟏,

𝝁𝟐,…𝝁𝒌 taking values in the range 𝟏 ≤ 𝝁𝟏, 𝝁𝟐, … , 𝝁𝒌 ≤ 𝑵! . The label

𝝁𝒌 identifies the permutation of the a priori unordered set �̃�[ �⃗⃗⃗�(𝒌−𝟏)]

(𝒌−𝟏)

the components of which are the N zeros of the polynomial of the

(k-1)-th generation with indices 𝝁𝟏, 𝝁𝟐, … , 𝝁𝒌−𝟏, i. e. the N zeros of

𝒑𝑵

(�⃗⃗⃗�(𝒌−𝟏);𝒌−𝟏)(𝒛; �⃗⃗⃗�

[�⃗⃗⃗�(𝒌−𝟏)]

(𝒌−𝟏) ; �̃�

[�⃗⃗⃗�(𝒌−𝟏)]

(𝒌−𝟏)) .


Question: Why introduce this notion of generations of monic polynomials? Reply: Why not? And note the possibility mentioned above to generate many solvable dynamical systems---including many-body problems of Newtonian type (“accelerations equal forces”)---associated with such polynomials. Question to the audience: is this notion of generations of polynomials new? Has it already been investigated?


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