Symmetric Determinantal Representations in Characteristic 2

TL;DR

This paper investigates symmetric determinantal representations (SDR) of multivariate polynomials in characteristic 2, providing conditions for existence, non-existence results, and algorithms for multilinear polynomials, with extensions to alternating representations.

Contribution

It offers new necessary and sufficient conditions for SDR existence in characteristic 2, including algorithms for testing factorization and constructing SDRs, and addresses the case of alternating representations.

Findings

Some polynomials have no SDR in characteristic 2.

Existence of SDR for multilinear polynomials relates to factorization in quotient rings.

Polynomial-time algorithms are developed for factorization and SDR construction.

Abstract

This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that P=det(M), and where each entry of M is either a constant or a variable. We first give some sufficient conditions for a polynomial to have an SDR. We then give a non-trivial necessary condition, which implies that some polynomials have no SDR, answering a question of Grenet et al. A large part of the paper is then devoted to the case of multilinear polynomials. We prove that the existence of an SDR for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings. We develop some algorithms to test the factorizability in these rings and use them to find SDRs when they exist. Altogether, this gives us polynomial-time algorithms to factorize the polynomials in the quotient rings and to build SDRs. We conclude by describing the case of Alternating Determinantal Representations in any characteristic.