Estimating the Number Of Roots of Trinomials over Finite Fields

Abstract

We show that univariate trinomials $x^n+a{x}^s+b\in {\mathbb{F}}_q[x]$ can have at most $\delta \lfloor \frac{1}{2}+\sqrt{\frac{q-1}{\delta }}\rfloor$ distinct roots in ${\mathbb{F}}_q$, where $\delta =\gcd (n,s,q-1)$. We also derive explicit trinomials having $\sqrt{q}$ roots in ${\mathbb{F}}_q$ when $q$ is square and $\delta =1$, thus showing that our bound is tight for an infinite family of finite fields and trinomials. Furthermore, we present the results of a large-scale computation which suggest that an $O(\delta \log q)$ upper bound may be possible for the special case where $q$ is prime. Finally, we give a conjecture (along with some accompanying computational and theoretical support) that, if true, would imply such a bound.