### 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 language | English (US) |
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Title of host publication | Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings |

Publisher | Springer-Verlag |

Pages | 482-491 |

Number of pages | 10 |

ISBN (Print) | 9783319964171 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 6th International Conference on Mathematical Software, ICMS 2018 - South Bend, United States Duration: Jul 24 2018 → Jul 27 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10931 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Conference on Mathematical Software, ICMS 2018 |
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Country | United States |

City | South Bend |

Period | 7/24/18 → 7/27/18 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - On μ -Symmetric Polynomials and D-Plus

AU - Yang, Jing

AU - Yap, Chee

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85050603729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050603729&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-96418-8_57

DO - 10.1007/978-3-319-96418-8_57

M3 - Conference contribution

AN - SCOPUS:85050603729

SN - 9783319964171

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 482

EP - 491

BT - Mathematical Software – ICMS 2018 - 6th International Conference, Proceedings

PB - Springer-Verlag

ER -