On the factorization of non-commutative polynomials (in free associative algebras)

Abstract

By viewing non-commutative polynomials, that is, elements in free associative algebras, in terms of linear representations, we generalize Horner's rule to the non-commutative (multivariate) setting. We introduce the concept of Horner systems (which has parallels to that of companion matrices), discuss their construction and show how they enable the efficient evaluation of non-commutative polynomials by matrices.

A Factorization Theory for Some Free Fields

“…In general however, it is necessary to minimize a given system. For a polynomial ALS this is discussed in [22,Section 2.2], for the general case we refer to [20].…”

“…The following definition is slightly adapted to avoid confusion with other transformation matrices for the factorization, formulated independent of a given admissible linear system. Definition 2.14 (Polynomial ALS and Transformation [22,Definition 24]). An ALS A = (u, A, v) of dimension n with system matrix A = (a ij ) for a non-zero…”

A standard form in (some) free fields: How to construct minimal linear representations

“…For details we refer to [2,Chapter 7] or [10,Section 6.4]. Since this work is only one part in a series, further information and references can be found in [11] (linear word problem, minimal inverse), [12] (polynomial factorization) and [13] (general factorization theory).…”

“…The intention of this paper is to be independent of the other three papers in this series (about the free field) as far as possible and leave it to the reader, for example, to interpret a standard form of the inverse of a polynomial as its factorization (into irreducible elements). Although the idea for minimizing "polynomial" linear representations is similar to that in Section 4, [12,Algorithm 32] is only a very special case of Algorithm 4.14 and the "refinement" in the former case is trivial.…”

“…The element f is then the first component of the (unique) solution vector = − s A v 1 . An ALS is also written as [12] to derive a "better" standard form including knowledge about the factorization it turned out later that this is not that easy in the general case because of the necessity to distinguish several cases in the multiplication [13,Section 5]. The term "pre-standard ALS" (for polynomials) is now replaced by polynomial ALS (which is just a special form of a refined ALS, Section 3).…”