In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory to $\mu$-symmetric polynomials. We were motivated by a conjecture from Becker et al.~(ISSAC 2016) about the $\mu$-symmetry of a particular root function $D^+(\mu)$, called D-plus. To investigate this conjecture, it was desirable to have fast algorithms for checking if a given root function is $\mu$-symmetric. We designed three such algorithms: one based on Gr\"{o}bner 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.

中文翻译：

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关于μ对称多项式

在本文中，我们研究单变量多项式的根函数，其中根具有给定的多重结构 $\mu$。传统上，根函数是通过对称多项式理论研究的；我们将这个理论扩展到 $\mu$ 对称多项式。我们受到 Becker 等人（ISSAC 2016）关于特定根函数 $D^+(\mu)$（称为 D-plus）的 $\mu$ 对称性的猜想的启发。为了研究这个猜想，需要有快速算法来检查给定的根函数是否是 $\mu$-对称的。我们设计了三种这样的算法：一种基于 Gr\"{o}bner 基，另一种基于预处理和约简，第三种基于求解线性方程。我们在 Maple 中实现了它们，实验表明后两种算法明显更快比第一个。