Estimating the Number Of Roots of Trinomials over Finite Fields

TL;DR

This paper establishes a tight upper bound on the number of roots of univariate trinomials over finite fields, provides explicit examples achieving this bound, and explores potential improvements and conjectures for prime fields.

Contribution

It introduces a new upper bound for roots of trinomials over finite fields, constructs explicit examples reaching this bound, and proposes conjectures for tighter bounds in prime fields.

Findings

Bound of $oxed{ ext{delta} imes loor{rac{1}{2} + ext{sqrt}(rac{q-1}{ ext{delta}})}}$ roots for trinomials

Explicit trinomials with $ ext{sqrt}(q)$ roots when $q$ is square and $ ext{delta}=1$

Computational evidence suggesting an $O( ext{delta} imes ext{log} q)$ bound for prime fields

Abstract

We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $\delta \Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}{\delta}} \Big\rfloor$ 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.