A Presentation of the Gfan Software

Gfan [8] is a software package for computing Grobner fans and tropical varieties of polynomial ideals. The Grobner fan of an ideal I⊂ℚ[x1,...,xn] is a polyhedral complex defined in [9]. For a homogeneous ideal the Grobner 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 Grobner bases of I. In [3] the local basis change of Grobner bases was introduced. This method allows us to go from one Grobner basis in the fan to a neighboring one, giving an effective algorithm for computing the Grobner 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].