On μ -Symmetric Polynomials and D-Plus

Jing Yang, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We study functions of the roots of a univariate polynomial of degree n≥ 1 in which the roots have a given multiplicity structure μ, denoted by a partition of n. For this purpose, we introduce a theory of μ -symmetric polynomials which generalizes the classic theory of symmetric polynomials. We designed three algorithms for checking if a given root function is μ -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 μ -symmetry of a certain root function D+(μ) called D-plus. This conjecture is proved to be true. But prior to the proof, we studied the conjecture experimentally using our algorithms.

Original languageEnglish (US)
Title of host publicationMathematical Software – ICMS 2018 - 6th International Conference, Proceedings
PublisherSpringer-Verlag
Pages482-491
Number of pages10
ISBN (Print)9783319964171
DOIs
StatePublished - Jan 1 2018
Event6th International Conference on Mathematical Software, ICMS 2018 - South Bend, United States
Duration: Jul 24 2018Jul 27 2018

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10931 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other6th International Conference on Mathematical Software, ICMS 2018
CountryUnited States
CitySouth Bend
Period7/24/187/27/18

Fingerprint

Symmetric Polynomials
Polynomials
Roots
Linear equations
Univariate
Preprocessing
Linear equation
Multiplicity
Partition
Symmetry
Generalise
Polynomial
Experiments
Experiment

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Yang, J., & Yap, C. (2018). On μ -Symmetric Polynomials and D-Plus. In Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings (pp. 482-491). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10931 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-319-96418-8_57

On μ -Symmetric Polynomials and D-Plus. / Yang, Jing; Yap, Chee.

Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings. Springer-Verlag, 2018. p. 482-491 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10931 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Yang, J & Yap, C 2018, On μ -Symmetric Polynomials and D-Plus. in Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10931 LNCS, Springer-Verlag, pp. 482-491, 6th International Conference on Mathematical Software, ICMS 2018, South Bend, United States, 7/24/18. https://doi.org/10.1007/978-3-319-96418-8_57
Yang J, Yap C. On μ -Symmetric Polynomials and D-Plus. In Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings. Springer-Verlag. 2018. p. 482-491. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-96418-8_57
Yang, Jing ; Yap, Chee. / On μ -Symmetric Polynomials and D-Plus. Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings. Springer-Verlag, 2018. pp. 482-491 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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