On $\mu$-Symmetric Polynomials and D-Plus

Abstract

We study functions of the roots of a univariate polynomial of degree $n\ge 1$ in which the roots have a given multiplicity structure ${\varvec{\mu }}$, denoted by a partition of n. For this purpose, we introduce a theory of ${\varvec{\mu }}$-symmetric polynomials which generalizes the classic theory of symmetric polynomials. We designed three algorithms for checking if a given root function is ${\varvec{\mu }}$-symmetric: one based on Gröbner bases, another based on preprocessing and reduction, and the third based on solving linear equations. Experiments show that the latter two algorithms are significantly faster. We were originally motivated by a conjecture about the ${\varvec{\mu }}$-symmetry of a certain root function $D^+({\varvec{\mu }})$ called D-plus. This conjecture is proved to be true. But prior to the proof, we studied the conjecture experimentally using our algorithms.

Jing’s work is supported by the Special Fund for Guangxi Bagui Scholars (WBS 2014-01) and the Startup Foundation for Advanced Talents in Guangxi University for Nationalities (2015MDQD018).

Chee’s work is supported by Guangxi University for Nationalities and by NSF Grant # CCF-1564132.