The $3$-sparsity of $X^n-1$ over finite fields

Abstract

Let $q$ be a prime power and ${\mathbb{F}}_q$ the finite field with $q$ elements. For a positive integer $n$, the binomial $X^n-1\in {\mathbb{F}}_q[X]$ is said to be $3$-sparse over ${\mathbb{F}}_q$ if every irreducible factor of $X^n-1$ in ${\mathbb{F}}_q[X]$ is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $X^n-1$ is $3$-sparse over ${\mathbb{F}}_q$ when $q=2$ and $q=4$, and raised the open problem of whether, for any given $q$, there are only finitely many primes $p$ such that $X^p-1$ is $3$-sparse over ${\mathbb{F}}_q$. In this paper, if $q$ is a power of an odd prime $r$, we then establish that for any positive integer not divisible by $r$, $X^n-1$ is $3$-sparse over ${\mathbb{F}}_q$ if and only if $n=p_1^{e_1}\cdots p_s^{e_s}$ for some nonnegative integers $e_1,\dots ,e_s$, where $p_1,\dots ,p_s$ are distinct prime divisors of $q^2-1$. This resolves the problem posed by Oliveira and Reis for odd characteristic.

10 pages