In computer algebra, the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of cones that correspond to different monomial orders on that ideal. The concept was introduced by Mora and Robbiano in 1988.[1] The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison.[2] Gröbner fan is a base for the nowadays active field of tropical geometry. One implementation of the Gröbner fan is called Gfan,[3] based on an article of Fukuda, et al.[4] which is included in some computer algebra systems such as Singular,[5] Macaulay2,[6] and CoCoA.[7]

See also

References

  1. ↑ Mora, Teo; Robbiano, Lorenzo (1988). "The Gröbner fan of an ideal". Journal of Symbolic Computation. 6 (2–3): 183–208. doi:10.1016/S0747-7171(88)80042-7.
  2. ↑ Bayer, David; Morrison, Ian (1988). "Standard bases and geometric invariant theory I. Initial ideals and state polytopes". Journal of Symbolic Computation. 6 (2–3): 209–217. doi:10.1016/S0747-7171(88)80043-9.
  3. ↑ "Gfan". home.math.au.dk. Retrieved 2017-04-03.
  4. ↑ Fukuda, Komei; Jensen, Anders N.; Thomas, Rekha R. (2007). "Computing Gröbner fans". Mathematics of Computation. 76 (260): 2189–2212. arXiv:math/0509544. doi:10.1090/S0025-5718-07-01986-2. MR 2336291.
  5. ↑ "Online Manual - groebnerFan". /www.singular.uni-kl.de. Retrieved 2022-02-23.
  6. ↑ "gfan -- all reduced Groebner bases of a polynomial ideal". faculty.math.illinois.edu. Retrieved 2022-02-23.
  7. ↑ "GroebnerFanReducedGBases". cocoa.dima.unige.it. Retrieved 2022-02-23.


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