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

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.