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

ABSTRACT. Given a collection P = {p1(x1,..., x2k2),..., pk2(x1,..., x2k2)} of k2 commutative polynomials in 2k2 variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative polyno-mial p(X,Y) in two k × k matrix variables X and Y. Algorithms that will gener-ically determine whether the given family P has a non-commutative representation and that will produce such a representation if they exist are developed. These algo-rithms 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 non-commutative polynomials in two variables. 1