关于μ-对称多项式

摘要：

本文研究一元多项式的根的函数，其中根具有给定的重数结构$μ$。传统上，根函数的研究通过对称多项式的理论，我们将这一理论推广到$μ$ -对称多项式。我们的动机来自Becker等人的一个猜想。（ISSAC 2016）关于$μ$ -特定根函数$D^+(μ)$的对称性，称为D+。为了研究这个猜想，人们希望有快速的算法来检查一个给定的根函数是否是$μ$对称的。我们设计了三个这样的算法：一个基于格罗布纳基，另一个基于预处理和约简，第三个基于求解线性方程组。我们在Maple中实现了它们，实验表明后两种算法的速度明显快于第一种算法。

英文摘要：

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $μ$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory to $μ$-symmetric polynomials. We were motivated by a conjecture from Becker et al.~(ISSAC 2016) about the $μ$-symmetry of a particular root function $D^+(μ)$, called D-plus. To investigate this conjecture, it was desirable to have fast algorithms for checking if a given root function is $μ$-symmetric. We designed three such algorithms: one based on Gröbner bases, another based on preprocessing and reduction, and the third based on solving linear equations. We implemented them in Maple and experiments show that the latter two algorithms are significantly faster than the first.

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