Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

Gorbovickis, Igors

doi:10.1017/etds.2014.103

Cited by 11 publications

(24 citation statements)

References 9 publications

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“…Let A be the associated matrix of Z(2, 1) = Z( δ, α, 2, 1), the exact same argument as step (iv) shows det A = (α 1 − α 2 )D(α 1 , α 2 ) = 0. Hence we conclude ev 3,4…”

Section: Proof Ofmentioning

confidence: 74%

“…As a consequence of this lemma, we can finish the proof that dim Z(3, 3) = 1. Let ev 3 : W (f ) ∼ = Z(3, 3) → C be the evaluation map p(x) −→ p(α 3 ). It suffices to check ker ev 3 = 0.…”

Section: Proof Ofmentioning

confidence: 99%

“…By the same manner as step (iv), we can check dim Z(4, 5) = 2. Let ev 3,4 : W (f ) → C 2 be the evaluation map p(x) −→ (p(α 3 ), p(α 4 )). As in step (iii), we get ker(ev 3,4…”

Section: Proof Ofmentioning

confidence: 99%

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Spaces of Polynomials Related to Multiplier Maps

Yang¹

2019

Math Notes

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Let f (x) be a complex polynomial of degree n. We attach to f a C-vector space W (f ) that consists of complex polynomials p(x) of degree at most n − 2 such that f (x) divides f ′′ (x)p(x) − f ′ (x)p ′ (x). W (f ) originally appears in Yuri Zarhin's solution towards a problem of dynamics in one complex variable posed by Yu. S. Ilyashenko. In this paper, we show that W (f ) is nonvanishing if and only if q(x) 2 divides f (x) for some quadratic polynomial q(x). Then we prove W (f ) has dimension (n − 1) − (n 1 + n 2 + 2N 3 ) under certain conditions, where n i is the number of distinct roots of f with multiplicity i and N 3 is the number of distinct roots of f with multiplicity at least three. Definitions, notation, and statementsWe write C for the field of complex numbers, C[x] for the ring of one variable polynomials with complex coefficients, and P s ⊆ C[x] for the subspace of polynomials of degree at most s. All vector spaces we consider are over C unless otherwise stated. Throughout the paper fand N 3 is the number of distinct roots of f with multiplicity at least three.We are interested in the space W (f ) ⊆ P n−2 , for which every p(x) ∈ W (f ) satisfies the condition that p(x) divides f ′′ (x)p(x) − f ′ (x)p ′ (x). As a subspace of P n−2 , we wish to compute the dimension of W (f ) for various f (x). The following assertions are main results of this paper.(iii) If n 1 = r = 4, and f (x) has at least two distinct multiple roots, then dim[W (f )] = 2 > µ = 1.Note that by rewriting n asUsing the lower bound of dim[W (f )] in Theorem 1.1.(i), we deduce if W (f ) vanishes then r ≤ n 1 − 1. Combining with the above expression of r, we obtainSince n i are nonnegative integers, we deduce from the above inequality that (n 2 , n 3 ) ∈ {(0, 0), (0, 1), (1, 0)} and n i = 0 for all i ≥ 4. This condition is equivalent to saying that for all quadratic polynomials q(x) ∈ C[x], q(x) 2 does not divide f (x). So by taking the contrapositive, we prove the following corollary.Corollary 1.2. If there exists a quadratic q(x)In fact, Zarhin showed that [6, Theorem 1.5.(ii)] the nonvanishing condition of W (f ) in Corollary 1.2 is sufficient. Suppose f is monic with distinct roots α 1 , . . . , α n . We consider a holomorphic map M : P n → C n defined via f (x) −→ (f ′ (α 1 ), . . . , f ′ (α n )). At first, Zarhin used W (f ) to show [6, Theorem 1.1] that the rank of the tangent map dM is n − 1 at all points of P n . Based on these results, Zarhin proved [7, Theorem 1.6] that the multipliers of any n − 1 distinct periodic orbits, considered as multi-valued algebraic maps on P n , are algebraically independent over C under certain conditions. This question about algebraic independence was asked by Yu. S. Ilyashenko in the context of studying the Kupka-Smale property for volume-preserving polynomial automorphisms of C 2 . (see [1,3] for a detailed discussion).

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“…Let A be the associated matrix of Z(2, 1) = Z( δ, α, 2, 1), the exact same argument as step (iv) shows det A = (α 1 − α 2 )D(α 1 , α 2 ) = 0. Hence we conclude ev 3,4…”

Section: Proof Ofmentioning

confidence: 74%

Section: Proof Ofmentioning

confidence: 99%

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Spaces of Polynomials Related to Multiplier Maps

Yang¹

2019

Math Notes

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“…Furthermore, for n = 6 one of these purely real critical points lies exactly at c = 0. The latter suggests the following: given a which was obtained in [3]. Using this formula, we can check numerically whether c = 0 is a critical point of the multiplier map λ n for periods n > 8.…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

“…Using this parameterization he proved that this moduli space is isomorphic to C 2 . In the attempt to generalize this approach, it was observed by the second author [3] that the multipliers of any m − 1 distinct periodic orbits provide a local parameterization of the moduli space of degree m polynomials in a neighborhood of its generic point. It is then a natural question to describe the set of polynomials at which this local parameterization fails, that is, to describe the set of all critical points of the multiplier map, defined as the map which assigns to each degree m polynomial the (m − 1)-tuple of multipliers at the chosen periodic orbits.…”

Section: Introductionmentioning

confidence: 99%

Critical Points of the Multiplier Map for the Quadratic Family

Belova

Gorbovickis

2019

Experimental Mathematics

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The multiplier λn of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials pc(z) = z 2 + c. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λn up to period n = 10.

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The moduli space of polynomial maps and their fixed-point multipliers

Sugiyama¹

2017

Advances in Mathematics

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We consider the family MP d of affine conjugacy classes of polynomial maps of one complex variable with degree d ≥ 2, and study the map Φ d :to the set of fixed-point multipliers of f . We show that the local fiber structure of the map Φ d aroundλ ∈ Λ d is completely determined by certain two sets I(λ) and K(λ) which are subsets of the power set of {1, 2, . . . , d}. Moreover for anyλ ∈ Λ d , we give an algorithm for counting the number of elements of each fiber Φ −1 d λ only by using I(λ) and K(λ). It can be carried out in finitely many steps, and often by hand.

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Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

Abstract: We consider a space of complex polynomials of degree n ≥ 3 with n − 1 distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent over C.

Cited by 11 publications

References 9 publications

Spaces of Polynomials Related to Multiplier Maps

Spaces of Polynomials Related to Multiplier Maps

Critical Points of the Multiplier Map for the Quadratic Family

The moduli space of polynomial maps and their fixed-point multipliers

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