NON-COMMUTATIVE REPRESENTATIONS OF FAMILIES OF k² COMMUTATIVE POLYNOMIALS IN 2k² COMMUTING VARIABLES

Abstract

Given a collection [Formula: see text] of k 2 commutative polynomials in 2k 2 variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative (nc) polynomial p(X, Y) in two k × k matrix variables X and Y. In this paper, we develop algorithms that will generically determine whether the given family [Formula: see text] has a nc representation and will produce such a representation if it exists. In particular, we determine an open, dense subset of the space of nc polynomials in two variables that satisfies the following property: if a family [Formula: see text] of polynomials admits a nc representation in this subset, then our algorithms will determine this representation.

Locations

• International Journal of Algebra and Computation - View
• arXiv (Cornell University) - PDF