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

Minoru Itoh; Jimpei Shimoyoshi

doi:10.1016/j.jalgebra.2021.03.017

paper

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

arXiv:2104.00838 · doi:10.1016/j.jalgebra.2021.03.017

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.

4 pages; to appear in J. Algebra