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Erschienen in:
2018 | OriginalPaper | Buchkapitel
On \(\mu \)-Symmetric Polynomials and D-Plus
verfasst von : Jing Yang, Chee K. Yap
Erschienen in: Mathematical Software – ICMS 2018
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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.