Solving Homogeneous Linear Equations over Polynomial Semirings
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).
wreath product
identity problem
polynomial semiring
positive polynomial
Computing methodologies~Symbolic and algebraic manipulation
26:1-26:19
Regular Paper
The author acknowledges support from UKRI Frontier Research Grant EP/X033813/1.
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].
Ruiwen
Dong
Ruiwen Dong
Department of Computer Science, University of Oxford, UK
10.4230/LIPIcs.STACS.2023.26
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Ruiwen Dong
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