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

TL;DR

This paper classifies when the polynomial $X^n-1$ over finite fields of characteristic two is 3-sparse, meaning its irreducible factors have at most three nonzero terms, providing a detailed characterization including an exceptional family.

Contribution

It provides a complete classification of 3-sparsity for $X^n-1$ over finite fields of characteristic two, including an explicit description of an exceptional family.

Findings

$X^n-1$ is 3-sparse if and only if $ ad(m) mid q^2-1$ and certain conditions hold.

The paper identifies an exceptional family involving powers of 7 with specific orbit conditions.

The classification includes a maximal 7-adic orbit condition for 3-sparsity.

Abstract

Let $q$ be a power of $2$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if every monic irreducible factor of $X^n-1$ over $\mathbb{F}_q$ has at most three nonzero terms. This corrected version gives the characteristic-two classification. Writing $n=2^\lambda m$ with $m$ odd, $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ if and only if either $\rad(m)\mid q^2-1$, or $q=2^e$, $3\nmid e$, and $m$ lies in the exceptional $7$-family \[ m=7^A s_0, \quad A\ge1, \quad (s_0,7)=1, \quad \rad(s_0)\mid q-1, \quad 3\nmid s_0/\gcd(s_0,q-1), \] with the additional maximal $7$-adic orbit condition $\ord_{7^a}(q)=3\cdot7^{a-1}$ for $1\le a\le A$. The latter condition is equivalent to $A=1$ or $7\nmid e$. This condition is necessary; for example, $X^{49}-1$ is not $3$-sparse over $\mathbb{F}_{128}$.