Conversion of the permanent into the determinant

Abstract

Let A A be an n n -square ( 0 , 1 ) (0,1) -matrix with positive permanent. It is shown that if the permanent of A A can be converted into a determinant by affixing ± \pm signs to the elements of A A then A A has at most ( n 2 + 3 n − 2 ) / 2 ({n^2} + 3n - 2)/2 positive entries. Corollaries of this result are given.