Non-Commutative Representations of families of K2 Commutative polynomials in 2K2 Commuting Variables

Given a collection P = {p1(x1, . . . , x2k2), . . . , pk2(x1, . . . , x2k2)} of k commutative polynomials in 2k variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative polynomial p(X,Y ) in two k × 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 if they exist are developed. These algorithms will determine a non-commutative representation for families P that admit a non-commutative representation in an open, dense subset of the vector space of noncommutative polynomials in two variables