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

Abstract: Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n - 1 \in \mathbb{F}_q[X]$ is termed $3$-sparse over $\mathbb{F}_q$ if all its irreducible factors in $\mathbb{F}_q[X]$ are either binomials or trinomials. 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 $q = 4$. Recently, the author provided a complete characterization for odd $q$. This paper extends the investigation to finite fields of even characteristic, fully determining all $n$ such that $X^n - 1$ is 3-sparse over $\mathbb{F}_q$ for even $q$. This work resolves two open problems posed by Oliveira and Reis for even characteristic case.