The Moduli Space of Rational Maps and Surjectivity of Multiplier Representation

Fujimura, Masayo

doi:10.1007/bf03321649

Cited by 12 publications

(24 citation statements)

References 6 publications

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“…For d ≥ 4, Fujimura and Nishizawa have long studied the map Φ d in their series of papers such as [16], [3] and [4]. Especially their achievement is summarized in Fujimura's paper [4], which includes the following:…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…• She showed in [4,Theorem 6] that if I(λ) is empty, then # Φ −1 d λ = (d − 2)! holds counted with multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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

Sugiyama¹

2017

Advances in Mathematics

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“…For rational maps of P 1 , relation (1.1) is the only one, as remarked early on by Fatou [Fat19,p. 168] (see also [Mil06, and the first part of the proof of theorem 1 in [Fuj07]).…”

Section: Introductionmentioning

confidence: 99%

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

Guillot

Ramírez

2019

Comput. Methods Funct. Theory

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This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of P 2 and quadratic homogeneous vector fields on C 3 , the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on C 3 having exclusively single-valued solutions.

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“…In the case of polynomials, the set of multipliers, or indices, at fixed points gives an interesting system of parameters on the moduli space of polynomials. For the details, see [3] and [4].…”

Section: Let Ratmentioning

confidence: 99%