Computing Gröbner fans

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.