Codes and Expansions (CodEx) Seminar

Nathaniel Johnston (Mount Allison University):
A New Formula for the Determinant

We present a new explicit formula for the determinant that contains significantly fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the rank of the n-by-n determinant tensor is no larger than the n-th Bell number, which is much smaller than the previously best-known upper bounds. Over fields of non-zero characteristic we obtain even tighter bounds, and in fields of characteristic 2 we obtain a formula for the permanent that has fewer terms than Ryser’s formula.