On the Polya permanent problem over finite fields

TL;DR

This paper investigates the transformations between permanent and determinant over finite fields, establishing limitations for bijective maps and providing examples of non-bijective and bijective maps, along with probabilistic estimates.

Contribution

It proves that no bijective map can transform permanent into determinant over large finite fields and offers new examples and probabilistic methods related to these functions.

Findings

No bijective map transforms permanent into determinant over large finite fields.

Existence of non-bijective maps transforming permanent into determinant over arbitrary fields.

Probabilistic estimates for the values of permanent and determinant over finite fields.

Abstract

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