Solving Homogeneous Linear Equations over Polynomial Semirings

Author Ruiwen Dong

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


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)


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
  • wreath product
  • identity problem
  • polynomial semiring
  • positive polynomial


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  1. Emil Artin. Über die zerlegung definiter funktionen in quadrate. In Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, volume 5, pages 100-115. Springer, 1927. Google Scholar
  2. William Ross Ashby. Automata Studies: Annals of Mathematics Studies. Number 34. Princeton University Press, 1956. Google Scholar
  3. François Baccelli, Guy Cohen, Geert Jan Olsder, and Jean-Pierre Quadrat. Synchronization and linearity: an algebra for discrete event systems. John Wiley & Sons Ltd, 1992. Google Scholar
  4. Paul C. Bell, Mika Hirvensalo, and Igor Potapov. The identity problem for matrix semigroups in SL₂(Z) is NP-complete. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 187-206. SIAM, 2017. Google Scholar
  5. Paul C. Bell and Igor Potapov. On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups. International Journal of Foundations of Computer Science, 21(06):963-978, 2010. Google Scholar
  6. Jan A. Bergstra and Jan Willem Klop. The algebra of recursively defined processes and the algebra of regular processes. In International Colloquium on Automata, Languages, and Programming, pages 82-94. Springer, 1984. Google Scholar
  7. Michaël Cadilhac, Dmitry Chistikov, and Georg Zetzsche. Rational subsets of Baumslag-Solitar groups. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 116:1-116:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  8. George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In Automata theory and formal languages, pages 134-183. Springer, 1975. Google Scholar
  9. Louis Dale. Monic and monic free ideals in a polynomial semiring. Proceedings of the American Mathematical Society, 56(1):45-50, 1976. Google Scholar
  10. Ruiwen Dong. On the identity problem and the group problem for subsemigroups of unipotent matrix groups, 2022. Submitted. URL:
  11. Samuel Eilenberg. Automata, languages, and machines. Academic press, 1974. Google Scholar
  12. Jonathan S. Golan. Semirings and affine equations over them: theory and applications, volume 556. Springer Science & Business Media, 2013. Google Scholar
  13. Rostislav I. Grigorchuk and Andrzej Żuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geometriae Dedicata, 87(1):209-244, 2001. Google Scholar
  14. Godfrey H. Hardy, John E. Littlewood, and G. Pólya. Inequalities. Cambridge University Press, Cambridge, 1952. Google Scholar
  15. Äbdelilah Kandri-Rody and Deepak Kapur. Computing a Gröbner basis of a polynomial ideal over a euclidean domain. Journal of symbolic computation, 6(1):37-57, 1988. Google Scholar
  16. Ravindran Kannan. Solving systems of linear equations over polynomials. Theoretical Computer Science, 39:69-88, 1985. Google Scholar
  17. Kenneth Krohn and John Rhodes. Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Transactions of the American Mathematical Society, 116:450-464, 1965. Google Scholar
  18. Markus Lohrey, Benjamin Steinberg, and Georg Zetzsche. Rational subsets and submonoids of wreath products. Information and Computation, 243:191-204, 2015. Google Scholar
  19. Wilhelm Magnus. On a theorem of Marshall Hall. Annals of Mathematics, pages 764-768, 1939. Google Scholar
  20. A. Markov. On certain insoluble problems concerning matrices. Doklady Akad. Nauk SSSR, 57(6):539-542, 1947. Google Scholar
  21. K. A. Mikhailova. The occurrence problem for direct products of groups. Matematicheskii Sbornik, 112(2):241-251, 1966. Google Scholar
  22. Paliath Narendran. Solving linear equations over polynomial semirings. In Proceedings 11th Annual IEEE Symposium on Logic in Computer Science, pages 466-472. IEEE, 1996. Google Scholar
  23. Jean-Eric Pin. Tropical Semirings. In J. Gunawardena, editor, Idempotency (Bristol, 1994), Publ. Newton Inst. 11, pages 50-69. Cambridge Univ. Press, Cambridge, 1998. URL:
  24. Alexander Prestel. Lectures on Formally Real Fields, volume 1093 of Lecture Notes in Mathematics. Springer, 2007. Google Scholar
  25. Alexander Prestel and Charles Delzell. Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra. Springer Science & Business Media, 2013. Google Scholar
  26. N. S. Romanovskii. Some algorithmic problems for solvable groups. Algebra and Logic, 13(1):13-16, 1974. Google Scholar
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