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

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 ${\mathbb{F}}_{128}$.

Corrected version. The exceptional characteristic-two family for prime 7 is revised. The implication 7 not | (q^2-1) => ord_{7^k}(q)=3*7^{k-1} fails for higher powers. Corrected: ord_{7^a}(q)=3*7^{a-1} for 1<=a<=A. Counterexample q=128, n=49. TLMS notified