Periodic points of polynomials over finite fields
Derek Garton · Transactions of the American Mathematical Society · 2022
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 d