Non-Commutative Representations of Families of k^2 Commutative Polynomials in 2k^2 Commuting Variables

TL;DR

This paper develops algorithms to determine and construct non-commutative matrix representations for families of commutative polynomials, enabling condensed non-commutative descriptions in algebraic systems.

Contribution

The paper introduces algorithms that can generically identify and produce non-commutative representations for polynomial families in a dense subset of the polynomial space.

Findings

Algorithms successfully determine non-commutative representations for certain polynomial families.

Representation exists for a dense subset of polynomial families in the space.

Provides a constructive method for non-commutative polynomial representation.

Abstract

Given a collection P of k^2 commutative polynomials in 2k^2 commutative variables, the objective is to find a condensed representation of these polynomials in terms of a single non-commutative polynomial p(X,Y) in two k x k matrix variables X and Y. Algorithms that will generically determine whether the given family P has a non-commutative representation and that will produce such a representation are developed. These algorithms will determine a non-commutative representation for families P that admit a a non-commutative representation in an open, dense subset of the vector space of non-commutative polynomials in two variables.