Periodic points of rational functions over finite fields
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913622139076608 |
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| author | Garton, Derek |
| author_facet | Garton, Derek |
| contents | For $q$ a prime power and $ϕ$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,ϕ)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $ϕ$. And if $d$ is a positive integer, let $Q_d$ be the set of prime powers coprime to $d!$ and let $\mathcal{P}(d,q)$ be the expected value of $p(q,ϕ)$ as $ϕ$ ranges over rational functions with coefficients in $\mathbb{F}_q$ of degree $d$. We prove that if $d$ is a positive integer no less than $2$, then $\mathcal{P}(d,q)$ tends to 0 as $q$ increases in $Q_d$. This theorem generalizes our previous work, which held only for quadratic polynomials, and only in fixed characteristic. To deduce this result, we prove a uniformity theorem on specializations of dynamical systems of rational functions with coefficients in certain finitely-generated algebras over residually finite Dedekind domains. This specialization theorem generalizes our previous work, which held only for algebras of dimension one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_13281 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Periodic points of rational functions over finite fields Garton, Derek Number Theory Dynamical Systems 37P05 (Primary), 37P25, 37P35, 11T06, 13B05 (Secondary) For $q$ a prime power and $ϕ$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,ϕ)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $ϕ$. And if $d$ is a positive integer, let $Q_d$ be the set of prime powers coprime to $d!$ and let $\mathcal{P}(d,q)$ be the expected value of $p(q,ϕ)$ as $ϕ$ ranges over rational functions with coefficients in $\mathbb{F}_q$ of degree $d$. We prove that if $d$ is a positive integer no less than $2$, then $\mathcal{P}(d,q)$ tends to 0 as $q$ increases in $Q_d$. This theorem generalizes our previous work, which held only for quadratic polynomials, and only in fixed characteristic. To deduce this result, we prove a uniformity theorem on specializations of dynamical systems of rational functions with coefficients in certain finitely-generated algebras over residually finite Dedekind domains. This specialization theorem generalizes our previous work, which held only for algebras of dimension one. |
| title | Periodic points of rational functions over finite fields |
| topic | Number Theory Dynamical Systems 37P05 (Primary), 37P25, 37P35, 11T06, 13B05 (Secondary) |
| url | https://arxiv.org/abs/2208.13281 |