Computing Gröbner fans
TL;DR: It is shown that the cells of a Grobner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memory-less reverse search procedure.
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Abstract: This paper presents algorithms for computing the Grobner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Grobner bases of the ideal. Our algorithms are based on a uniform definition of the Grobner fan that applies to both homogeneous and non-homogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Grobner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memory-less reverse search procedure. The significance of this follows from the fact that Grobner fans are not always normal fans of polyhedra, in which case reverse search applies automatically. Computational results using our implementation of these algorithms in the software package Gfan are included.
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Citations
Beyond Grobner Bases: Basis Selection for Minimal Solvers
Viktor Larsson,Magnus Oskarsson,Kalle Åström,Alge Wallis,Tomas Pajdla,Zuzana Kukelova +5 more
- 18 Jun 2018
TL;DR: In this article, the action matrix method is used to make polynomial solvers based on Grobner bases faster, by careful selection of the monomial bases, which leads to more efficient solvers in many cases.
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Computing the GIT-fan
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Torical modification of Newton non-degenerate ideals
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References
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Introduction to Toric Varieties.
William Fulton
- 12 Jul 1993
TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
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Gröbner bases and convex polytopes
Bernd Sturmfels
- 14 Dec 1995
TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
1.9K
Reverse search for enumeration
David Avis,Komei Fukuda +1 more
TL;DR: This paper develops the reverse search technique in a general framework and shows its broader applications to various problems in operations research, combinatorics, and geometry, and proposes new algorithms for listing.
899
Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering
TL;DR: The lexicographical GroBner basis can be obtained by applying this algorithm after a total degree Grobner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchberger's algorithm.
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