• DOI:10.48550/arXiv.2209.13347
• Corpus ID: 252544982

Solving homogeneous linear equations over polynomial semirings

@article{Dong2022SolvingHL,
  title={Solving homogeneous linear equations over polynomial semirings},
  author={Rui-Tao Dong},
  journal={ArXiv},
  year={2022},
  volume={abs/2209.13347}
}

• Rui-Tao Dong
• Published 27 September 2022
• Mathematics
• ArXiv

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 o 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 o Z is decidable when elements of the semigroup generator have the form ( y, ± 1). 2012 ACM Subject Classification Computing methodologies → Symbolic and algebraic manipulation Keywords Acknowledgements The author would like to thank Markus Schweighofer for discussion and for pointing out the references [24] and [25].