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Solving Homogeneous Linear Equations over Polynomial Semirings

Author Ruiwen Dong

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LIPIcs.STACS.2023.26.pdf
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Author Details

Ruiwen Dong
  • Department of Computer Science, University of Oxford, UK

Funding

The author acknowledges support from UKRI Frontier Research Grant EP/X033813/1.

Acknowledgements

The author would like to thank Markus Schweighofer for useful discussions and feedback and for pointing out the references [Prestel, 2007] and [Prestel and Delzell, 2013].

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Ruiwen Dong. Solving Homogeneous Linear Equations over Polynomial Semirings. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.26

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).

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic manipulation
Keywords
  • wreath product
  • identity problem
  • polynomial semiring
  • positive polynomial

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