Derek Garton
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View article: Periodic points of rational functions over finite fields
Periodic points of rational functions over finite fields Open
For a prime power and a rational function with coefficients in , let be the proportion of that is periodic with respect to . Furthermore, if is a positive integer, let be the set of prime powers coprime to and let be the expected value of …
View article: Preperiodic points of polynomial dynamical systems over finite fields
Preperiodic points of polynomial dynamical systems over finite fields Open
For a prime p, positive integers [Formula: see text], and a polynomial f with coefficients in [Formula: see text], let [Formula: see text]. As n varies, the [Formula: see text] partition the set of strictly preperiodic points of the dynami…
View article: Periodic points of rational functions over finite fields
Periodic points of rational functions over finite fields Open
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 …
View article: Periodic points of polynomials over finite fields
Periodic points of polynomials over finite fields Open
Fix an odd prime . If is a positive integer and is a polynomial with coefficients in , let be the proportion of that is periodic with respect to . We show that as increases, the expected value of , as ranges over quadratic polynomial…
View article: Dynamically distinguishing polynomials
Dynamically distinguishing polynomials Open
A polynomial with integer coefficients yields a family of dynamical systems\nindexed by primes as follows: for any prime $p$, reduce its coefficients mod\n$p$ and consider its action on the field $\\mathbb{F}_p$. We say a subset of\n$\\mat…
View article: Random matrices, the Cohen–Lenstra heuristics, and roots of unity
Random matrices, the Cohen–Lenstra heuristics, and roots of unity Open
The Cohen-Lenstra-Martinet heuristics predict the frequency with which a\nfixed finite abelian group appears as an ideal class group of an extension of\nnumber fields, for certain sets of extensions of a base field. Recently, Malle\nfound …