论文标题
在MU对称多项式上
On mu-Symmetric Polynomials
论文作者
论文摘要
在本文中，我们研究了单变量多项式的根的功能，其中根部具有给定的多重性结构$μ$。传统上，通过对称多项式理论研究了根函数。我们将该理论扩展到$ $ $ - 符合性的多项式。我们是由Becker等人（ISSAC 2016）的猜想激励的，涉及特定根函数$ d^+（μ）$的$ $ $ symmetry，称为d-plus。为了研究这种猜想，希望具有快速算法来检查给定的根函数是否为$μ$ - 符合物。我们设计了三种这样的算法：一种基于Gröbner基础，另一个基于预处理和还原，第三个基于求解线性方程。我们在枫木中实现了它们，实验表明后两种算法明显比第一个算法快得多。
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.
