The Cycle Structure of Unicritical Polynomials

Bridy, Andrew; Garton, Derek

doi:10.1093/imrn/rny232

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…As an example of the utility of this approach, we recall that Pollard's famous "rho" method of factorization [Pol75] relies on an aspect of the purported randomness of the family of dynamical systems associated to R = Z and f = X 2 + 1. For recent work on this approach to studying randomness, see [JKMT16,BG17,BG20,Juu]. To ease notation, for any commutative ring R, we write P R for the nonzero prime ideals of R; moreover, if R is residually finite, then for any p ∈ P R , we write N (p) for [R] p .…”

Section: Introductionmentioning

confidence: 99%

Periodic points of polynomials over finite fields

Garton

2021

Preprint

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Fix an odd prime p. If r is a positive integer and f polynomial with coefficients in F p r , let P p,r (f ) be the proportion of P 1 (F p r ) that is periodic with respect to f . We show that as r increases, the expected value of P p,r (f ), as f ranges over quadratic polynomials, is less than 22 (log log p r ). This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.

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Section: Introductionmentioning

confidence: 99%

Periodic points of polynomials over finite fields

Garton

2021

Preprint

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Functional graphs of families of quadratic polynomials

et al. 2023

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We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.

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Periodic points of polynomials over finite fields

Garton

2022

Trans. Amer. Math. Soc.

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Fix an odd prime p p . If r r is a positive integer and f f is a polynomial with coefficients in F p r \mathbb {F}_{p^r} , let P p , r ( f ) P_{p,r}(f) be the proportion of P 1 ( F p r ) \mathbb {P}^1\left (\mathbb {F}_{p^r}\right ) that is periodic with respect to f f . We show that as r r increases, the expected value of P p , r ( f ) P_{p,r}(f) , as f f ranges over quadratic polynomials, is less than 22 / ( log ⁡ log ⁡ p r ) 22/\left (\log {\log {p^r}}\right ) . This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.

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The Cycle Structure of Unicritical Polynomials

Cited by 4 publications

References 33 publications

Periodic points of polynomials over finite fields

Periodic points of polynomials over finite fields

Functional graphs of families of quadratic polynomials

Periodic points of polynomials over finite fields

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