On the Polya permanent problem over finite fields

Let FF be a finite field of characteristic different from 2. We show that no bijective map transforms the permanent into the determinant when the cardinality of FF is sufficiently large. We also give an example of a non-bijective map when FF is arbitrary and an example of a bijective map when FF is infinite which do transform the permanent into the determinant. The technique developed allows us to estimate the probability of the permanent and the determinant of matrices over finite fields having a given value. Our results are also true over finite rings without zero divisors.