A Polynomial-time Algorithm for Computing the Permanent in GF(3^q)

TL;DR

This paper introduces a polynomial-time algorithm for computing the permanent over fields of characteristic 3, utilizing classical matrix identities and novel limit techniques, with implications for complexity class equivalences.

Contribution

It presents the first polynomial-time algorithm for the permanent in fields of characteristic 3, extending classical methods with new limit-based techniques.

Findings

Permanent computability in GF(3^q) is polynomial-time

Establishes a link between permanent complexity and P vs NP

Uses classical identities and novel limit methods

Abstract

A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and their generalizations of various types. Classical theorems on the permanent such as the Binet-Minc identity and Borchadt's formula are widely applied, while a special new technique involving the notion of limit re-defined for fields of finite characteristics and corresponding computational methods was developed in order to deal with a number of polynomial-time reductions. All the constructions preserve a strictly algebraic nature ignoring the structure of the basic field, while applying its infinite extensions for calculating limits. A natural corollary of the polynomial-time computability of the permanent in a field of a characteristic different from 2 is the non-uniform equality between the complexity classes P and NP what is equivalent to RP=NP (Ref. [1]).