An application of the Hasse--Weil bound to rational functions over finite fields
Volume 195 / 2020
Acta Arithmetica 195 (2020), 207-216
MSC: Primary 11T06; Secondary 11R58, 14H05.
DOI: 10.4064/aa190701-5-12
Published online: 13 May 2020
Abstract
We use the Aubry–Perret bound for singular curves, a generalization of the Hasse–Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in \Bbb F_q(X) \setminus \Bbb F_q$ be such that $q$ is sufficiently large relative to $\deg f$ and $\deg g$, $f(\Bbb F_q)\subset g(\Bbb F_q\cup \{\infty \})$, and for “most” $a\in \Bbb F_q\cup \{\infty \}$, $|\{x\in \Bbb F_q:g(x)=g(a)\}| \gt (\deg g)/2$. Then there exists $h(X)\in \Bbb F_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included.
