# Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions

## TL;DR

This paper explores efficient evaluation methods for noncommutative polynomials using tensor and Waring decompositions, aiming to reduce matrix multiplications and improve computational speed.

## Contribution

It introduces a noncommutative Waring decomposition framework, compares it with classical approaches, and provides methods for computing these decompositions to enhance evaluation efficiency.

## Key findings

- Decomposition reduces matrix multiplications needed for evaluation.
- Comparison shows noncommutative polynomials differ from commutative ones in decomposability.
- Proposed methods improve evaluation speed for generic noncommutative polynomials.

## Abstract

This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial must be evaluated on many matrix tuples. In pursuit of this goal we examine a noncommutative analog of the classical Waring problem and various related decompositions. For example, we consider a "Waring decomposition" in which each product of linear terms is actually a power of a single linear NC polynomial or more generally a power of a homogeneous NC polynomial. We describe how NC polynomials compare to commutative ones with regard to these decompositions, describe a method for computing the NC decompositions and compare the effect of various decompositions on the speed of evaluation of generic NC polynomials.