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.

摘要

本文针对评估p的目标分析了非换向（NC）多项式p的Waring类型分解高效地处理矩阵元组。这样的分解可以减少评估非可交换多项式所需的矩阵乘法次数，并且在必须对多个矩阵元组评估一个多项式时非常有用。为了实现这一目标，我们研究了经典Waring问题和各种相关分解的非可交换类比。例如，我们考虑“战争分解”，其中线性项的每个乘积实际上是单个线性NC多项式的幂或更通常是齐次NC多项式的幂。我们将针对这些分解描述N​​C多项式与可交换多项式的比较，描述一种计算NC分解的方法，并比较各种分解对通用NC多项式求值速度的影响。