Permanent has less zeros than determinant over finite fields
M. V. Budrevich, Alexander Emilevich Guterman · Contemporary mathematics - American Mathematical Society · 2012
Let F q \mathbb {F}_q be an arbitrary finite field of characteristic different from two. We show that the permanent function has less zeros than the determinant function for square matrices of an arbitrary size n > 2 n>2 over F q \mathbb {F}_q . As a consequence, we obtain the answer to the Pólya problem over F q \mathbb {F}_q by showing that there are no bijective transformations on matrices with entries from F q \mathbb {F}_q which map the permanent into the determinant.