Wendt's determinant and finiteness of prime characteristics for permutation binomials

Abstract

For $f(X)=X^r(X^s+a)\in\mathbb F_{p^m}[X]$, with $m,r,s\geqslant1$ and $a\in\mathbb F_{p^m}^\times$, put $g=\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\nmid g$, every such $p$ divides Wendt's determinant $W_g=\operatorname{Res}_X(X^g-1,(X+1)^g-1)$, which is nonzero. When $6\mid g$, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such $p>2g+1$. We determine the exact admissible characteristics for $1\leqslant g\leqslant10$, obtain a further divisibility condition over prime fields, and show that in prime fields $g=11$ occurs precisely when $p=23$.