Solving homogeneous linear equations over polynomial semirings
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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).
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