Gröbner fan

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.{{Cite journal|title=The Gröbner fan of an ideal |language=en|doi=10.1016/S0747-7171(88)80042-7|volume=6|issue=2–3 |journal=Journal of Symbolic Computation|pages=183–208 | last2 = Robbiano | first2 = Lorenzo | last1 = Mora | first1 = Teo|year=1988 |doi-access=free }}

The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison.{{Cite journal|title=Standard bases and geometric invariant theory I. Initial ideals and state polytopes|language=en|doi=10.1016/S0747-7171(88)80043-9|volume=6|issue=2–3|journal=Journal of Symbolic Computation|pages=209–217 | last1 = Bayer | first1 = David | last2 = Morrison | first2 = Ian|year=1988|doi-access=free}} Gröbner fan is a base for the nowadays active field of tropical geometry.

One implementation of the Gröbner fan is called Gfan,{{Cite web|url=http://home.math.au.dk/jensen/software/gfan/gfan.html|title=Gfan|website=home.math.au.dk|access-date=2017-04-03}} based on an article of Fukuda, et al.{{cite journal | last1 = Fukuda | first1 = Komei | author1-link = Komei Fukuda | last2 = Jensen | first2 = Anders N. | last3 = Thomas | first3 = Rekha R. | doi = 10.1090/S0025-5718-07-01986-2 | issue = 260 | journal = Mathematics of Computation | mr = 2336291 | pages = 2189–2212 | title = Computing Gröbner fans | volume = 76 | year = 2007| arxiv = math/0509544 }} which is included in some computer algebra systems such as Singular,{{Cite web|title=Online Manual - groebnerFan|url=https://www.singular.uni-kl.de/Manual/4-3-0/sing_2606.htm|access-date=2022-02-23|website=/www.singular.uni-kl.de}} Macaulay2,{{Cite web|title=gfan -- all reduced Groebner bases of a polynomial ideal|url=https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/gfanInterface/html/_gfan.html|access-date=2024-02-09|website=faculty.math.illinois.edu}} and CoCoA.{{Cite web|title=GroebnerFanReducedGBases|url=https://cocoa.dima.unige.it/cocoa/download/CoCoAManual/html/cmdGroebnerFanReducedGBases.html|access-date=2024-02-09|website=cocoa.dima.unige.it}}