Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

SAGBI and Gröbner Bases Detection (2404.16796v1)

Published 25 Apr 2024 in math.AC and math.AG

Abstract: We introduce a detection algorithm for SAGBI basis in polynomial rings, analogous to a Gr\"obner basis detection algorithm previously proposed by Gritzmann and Sturmfels. We also present two accompanying software packages named SagbiGbDetection for Macaulay2 and Julia. Both packages allow the user to find one or more term orders for which a set of input polynomials form either Gr\"obner basis for the ideal they generate or a SAGBI basis for the subalgebra. Additionally, we investigate the computational complexity of homogeneous SAGBI detection and apply our implementation to several novel examples.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (40)
  1. Julia: A fresh approach to numerical computing. SIAM review, 59(1):65–98, 2017.
  2. René Birkner. Polyhedra: a package for computations with convex polyhedral objects. Journal of Software for Algebra and Geometry, 1(1):11–15, 2009.
  3. Example codes of this manuscript. Available at https://github.com/elimashehu/SagbiGbDetection_Applications.git.
  4. SagbiGbDetection: . A Julia package available at https://github.com/V-Borovik/SagbiGbDetection.jl.git.
  5. SagbiGbDetection: A Macaulay2 package. Version 0.1. Available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages.
  6. Sagbi combinatorics of maximal minors and a sagbi algorithm. arXiv.2302.14345, 2023.
  7. Determinants, Gröbner Bases and Cohomology, volume XIII, Springer Monographs in Mathematics. Springer, 2022.
  8. Bruno Buchberger. Bruno buchberger’s phd thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Journal of Symbolic Computation, 41(3):475–511, 2006.
  9. Subalgebrabases in macaulay2. To appear in Journal of Software for Algebra and Geometry, 2024.
  10. Numerical homotopies from khovanskii bases. Mathematics of Computation, 92, 2020.
  11. Toric degenerations of grassmannians and schubert varieties from matching field tableaux. Journal of Algebra, 559:646–678, 2020.
  12. Universal grobner bases for maximal minors. International Mathematics Research Notices, 2015, 2013.
  13. Sullivant-Talaska ideal of the cyclic gaussian graphical model. arXiv.2308.05561, 2023.
  14. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. 2007.
  15. Singular 4-1-2—a computer algebra system for polynomial computations. http://www. singular. uni-kl. de, 2019.
  16. Algebraic varieties in quantum chemistry. arXiv.2308.05258, 2023.
  17. Coupled cluster theory: Toward an algebraic geometry formulation. SIAM Journal on Applied Algebra and Geometry, 8(1):138–188, 2024.
  18. Polymake: a framework for analyzing convex polytopes. In Polytopes—combinatorics and computation, pages 43–73. Springer, 2000.
  19. Macaulay2, a software system for research in algebraic geometry. Available at https://macaulay2.com/, 2020.
  20. Minkowski addition of polytopes: computational complexity and applications to Gröbner bases. SIAM J. Discrete Math., 6(2):246–269, 1993.
  21. Hyperdeterminantal relations among symmetric principal minors. Journal of Algebra, 316(2):634–648, 2007.
  22. Numerical schubert calculus. Journal of Symbolic Computation, 26(6):767–788, 1998.
  23. Polymake. jl: A new interface to polymake. In International Congress on Mathematical Software, pages 377–385. Springer, 2020.
  24. A completion procedure for computing a canonical basis for a k-subalgebra. In Erich Kaltofen and Stephen M. Watt, editors, Computers and Mathematics, pages 1–11, New York, NY, 1989. Springer US.
  25. K. Kaveh and C. Manon. Khovanskii bases, higher rank valuations, and tropical geometry. SIAM Journal on Applied Algebra and Geometry, 3.2:292–336, 2019.
  26. Newton-okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Annals of Mathematics, 176(2):925–978, 2012.
  27. Computational Commutative Algebra 2. 01 2005.
  28. Shigeru Kuroda. A new class of finitely generated polynomial subalgebras without finite sagbi bases. arXiv preprint arXiv:2110.08748, 2021.
  29. Calibration-free structure-from-motion with calibrated radial trifocal tensors. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part V 16, pages 382–399. Springer, 2020.
  30. Convex bodies associated to linear series. Annales scientifiques de l’École Normale Supérieure, 42(5):783–835, 2009.
  31. Igor Makhlin. Chain-order polytopes: toric degenerations, young tableaux and monomial bases. 2022.
  32. M. Michałek and B. Sturmfels. Invitation to nonlinear algebra, volume 211, Graduate Studies in Mathematics. American Mathematical Society, 2021.
  33. Christos H. Papadimitriou. On the complexity of integer programming. J. ACM, 28(4):765–768, 1981.
  34. Subalgebra bases. In Commutative Algebra: Proceedings of a Workshop held in Salvador, Brazil, Aug. 8–17, 1988, pages 61–87. Springer, 2006.
  35. Frank Sottile. Real solutions to equations from geometry. In University Lecture Series, 2006.
  36. The tropical grassmannian. Advances in Geometry, 4:389–411, 2003.
  37. Bernd Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, Berlin, Heidelberg, 1993.
  38. Bernd Sturmfels. Equations defining toric varieties. arXiv: Algebraic Geometry, 1996.
  39. Bernd Sturmfels. Grobner bases and convex polytopes, volume 8. American Mathematical Soc., 1996.
  40. Seth Sullivant. Gaussian conditional independence relations have no finite complete characterization. Journal of Pure and Applied Algebra, 213(8):1502–1506, 2009. Theoretical Effectivity and Practical Effectivity of Gröbner Bases.

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com