利用张量和非交换Waring分解计算非交换多项式
Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions
摘要：
本文分析了非交换（NC）多项式$p$的Waring型分解，目的是在矩阵元组上有效地计算$p$.这样的分解可以减少计算非交换多项式所需的矩阵乘法的数量，并且在必须在许多矩阵元组上计算单个多项式时是有价值的。在追求这一目标，我们研究了一个非交换模拟的经典华林问题和各种相关的分解。例如，我们考虑“华林分解”，其中线性项的每个乘积实际上是单个线性NC多项式的幂，或者更一般地是齐次NC多项式的幂。我们描述了如何NC多项式比较交换的关于这些分解，描述了一种方法，用于计算NC分解和比较各种分解的效果对通用NC多项式的评价速度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
