A condition for the existence of zero coefficients in the powers of the determinant polynomial

TL;DR

This paper characterizes when zero coefficients appear in powers of the determinant polynomial, establishing a condition that relates to prime numbers and extending Glynn's earlier results.

Contribution

It proves that the coefficients of the mth power of the determinant polynomial are all nonzero if and only if m equals p-1 for some prime p, for n ≥ 3.

Findings

Coefficients are all nonzero when m = p-1, with p prime.

The converse of Glynn's result is proven for n ≥ 3.

Elementary proof provided for the main result.

Abstract

We discuss the existence of zero coefficients in the powers of the determinant polynomial of order $n$. D. G. Glynn proved that the coefficients of the $m$th power of the determinant polynomial are all nonzero, if $m = p-1$ with a prime $p$. We show that the converse also holds, if $n \geq 3$. The proof is quite elementary.