Matrix coefficient realization theory of noncommutative rational functions

Abstract

Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas, from free real algebraic geometry to systems and control theory. If a noncommutative rational function is regular at the origin, it can be described by a linear object, called a realization. In this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rationas functions and is welladapted for studying matrix evaluations.Of special interest are the minimal realizations, which compensate the absence of a canonical form for noncommutative rational functions. The non-minimality of a realization is assessed by obstruction modules associated with it; they enable us to devise an efficient method for obtaining minimal realizations. With them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity. Using these results we determine concrete size bounds for rational identity testing, construct minimal symmetric realizations and prove an effective local-global principle for linear dependence of noncommutative rational functions.

On domains of noncommutative rational functions

“…In [Vol+], matrix coefficient realization theory is applied to extend Corollary 3.7 to arbitrary nc rational functions (i.e., those not necessarily defined at a scalar point) in terms of their Sylvester realizations. Thus every stable extended domain can be described as the invertibility set of a generalized monic pencil; see [Vol+,Corollary 5.9] for the precise statement.…”

Stable Noncommutative Polynomials and Their Determinantal Representations

“…We say that (3.1) is a (descriptor) realization of Ö of size δ; see [BGM05, Section 12] and [HMV06,Vol18]. In automata theory, such realizations are also called linear representations [BR11].…”

The free Grothendieck theorem

“…The free product of and is the set of all words , where , are non‐empty words. A much more detailed exposition can be found in . Definition If and has a non‐zero constant term , then the multiplicative inverse of , exists and is given by Let denote the algebra of rational series ; the smallest subalgebra of containing such that if and exists, then .…”

“…Lemma implies , the inverse of , is the unique solution to a proper algebraic polynomial that is ‐affine linear. The ‐affine linearity is reminiscent of realizations of nc rational functions, see . With this similarity to realizations in mind, we generalize the class of rational series (see Definition ) to a slightly larger class of formal power series that we call the hyporational series.…”