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Subalgebra and Khovanskii bases equivalence (2402.06057v1)

Published 8 Feb 2024 in math.AG and cs.SC

Abstract: The main results of this paper establish a partial correspondence between two previously-studied analogues of Groebner bases in the setting of algebras: namely, subalgebra (aka SAGBI) bases for quotients of polynomial rings and Khovanskii bases for valued algebras. We aim to bridge the gap between the concrete, computational aspects of the former and the more abstract theory of the latter. Our philosophy is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. We also discuss the computation of Newton-Okounkov bodies, illustrating how interpreting Khovanskii bases as subalgebra bases makes them more amenable to the existing computer algebra tools.

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References (17)
  1. CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
  2. D. Anderson. Okounkov bodies and toric degenerations. Mathematische Annalen, 356:1183–1202, 2013.
  3. Saturations of subalgebras, sagbi bases, and u-invariants. Journal of Symbolic Computation, 109:259–282, 2022.
  4. SAGBI combinatorics of maximal minors and a SAGBI algorithm. Journal of Symbolic Computation, 120, 2024.
  5. SubalgebraBases: Canonical subalgebra bases (aka SAGBI/Khovanskii bases). A Macaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages.
  6. Subalgebrabases in Macaulay2. Technical Report arXiv:2302.12473 [math.AC], arXiv, 2023.
  7. Numerical homotopies from Khovanskii bases. Mathematics of Computation, 92(343):2333–2353, 2023.
  8. Singular 4-3-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2023.
  9. Manfred Göbel. Computing bases for rings of permutation-invariant polynomials. Journal of Symbolic Computation, 19(4):285–291, 1995.
  10. Macaulay2, a software system for research in algebraic geometry. Available at http://www2.macaulay2.com.
  11. sagbi.lib. singular 4-3-2 library for computing SAGBI bases of a subalgebra. https://www.singular.uni-kl.de/Manual/4-3-0/sing_1556.htm, 2023.
  12. 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.
  13. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Annals of Mathematics, 176(2):925–978, 2012.
  14. Khovanskii bases, higher rank valuations, and tropical geometry. SIAM Journal on Applied Algebra and Geometry, 3(2):292–336, 2019.
  15. L. Robbiano and M. Sweedler. Subalgebra bases. In Winfried Bruns and Aron Simis, editors, Commutative Algebra, pages 61–87, Berlin, Heidelberg, 1990. Springer Berlin Heidelberg.
  16. Using sagbi bases to compute invariants. Journal of Pure and Applied Algebra, 139(1):285–302, 1999.
  17. Bernd Sturmfels. Gröbner Bases and Convex Polytopes, volume 8 of University lecture series. American Mathematical Society, Providence, RI, 1996.

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