License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2023.26
URN: urn:nbn:de:0030-drops-176784
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17678/
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Dong, Ruiwen

Solving Homogeneous Linear Equations over Polynomial Semirings

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LIPIcs-STACS-2023-26.pdf (0.8 MB)

Abstract

For a subset B of ℝ, denote by U(B) be the semiring of (univariate) polynomials in ℝ[X] that are strictly positive on B. Let ℕ[X] be the semiring of (univariate) polynomials with non-negative integer coefficients. We study solutions of homogeneous linear equations over the polynomial semirings U(B) and ℕ[X]. In particular, we prove local-global principles for solving single homogeneous linear equations over these semirings. We then show PTIME decidability of determining the existence of non-zero solutions over ℕ[X] of single homogeneous linear equations.
Our study of these polynomial semirings is largely motivated by several semigroup algorithmic problems in the wreath product ℤ≀ℤ. As an application of our results, we show that the Identity Problem (whether a given semigroup contains the neutral element?) and the Group Problem (whether a given semigroup is a group?) for finitely generated sub-semigroups of the wreath product ℤ≀ℤ is decidable when elements of the semigroup generator have the form (y, ±1).

BibTeX - Entry

@InProceedings{dong:LIPIcs.STACS.2023.26,
  author =	{Dong, Ruiwen},
  title =	{{Solving Homogeneous Linear Equations over Polynomial Semirings}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17678},
  URN =		{urn:nbn:de:0030-drops-176784},
  doi =		{10.4230/LIPIcs.STACS.2023.26},
  annote =	{Keywords: wreath product, identity problem, polynomial semiring, positive polynomial}
}

Keywords: wreath product, identity problem, polynomial semiring, positive polynomial
Collection: 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)
Issue Date: 2023
Date of publication: 03.03.2023

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