2026-07-24T08:01:24Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17678 2024-03-06T11:00:17Z ddc:004 open_access

Solving Homogeneous Linear Equations over Polynomial Semirings Dong, Ruiwen wreath product identity problem polynomial semiring positive polynomial 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). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ruiwen Dong 2023 Is Part Of LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2023.26 urn:nbn:de:0030-drops-176784 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.26 eng https://creativecommons.org/licenses/by/4.0/legalcode