Horner Systems: How to efficiently evaluate non-commutative polynomials (by matrices)

TL;DR

This paper generalizes Horner's rule to non-commutative polynomials using linear representations, introducing Horner systems for efficient matrix evaluation of such polynomials.

Contribution

It introduces Horner systems for non-commutative polynomials, providing a new method for their efficient evaluation via matrices.

Findings

Horner systems enable efficient polynomial evaluation.

The approach generalizes Horner's rule to multivariate non-commutative polynomials.

Construction of Horner systems parallels that of companion matrices.

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.