A Presentation of the Gfan Software

  • Anders N. Jensen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)


Gfan [8] is a software package for computing Gröbner fans and tropical varieties of polynomial ideals. The Gröbner fan of an ideal I ⊂ ℚ[x 1,...,x n ] is a polyhedral complex defined in [9]. For a homogeneous ideal the Gröbner fan is a complete fan and the normal fan of a polytope. Its cones are in bijection with the various initial ideals of I. In particular, the full dimensional cones are in bijection with the monomial initial ideals and thereby also in bijection with the reduced Gröbner bases of I. In [3] the local basis change of Gröbner bases was introduced. This method allows us to go from one Gröbner basis in the fan to a neighboring one, giving an effective algorithm for computing the Gröbner fan by traversing its maximal cones. The method can be refined by applying the reverse search technique [1]. This works even in the non-homogeneous case [5].


Polynomial Ideal Homogeneous Ideal Initial Ideal Tropical Variety Toric Ideal 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anders N. Jensen
    • 1
  1. 1.Department of Mathematical SciencesUniversity of AarhusÅrhusDenmark

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