2005/09/23 by Komei Fukuda, Anders Jensen, Anders N. Jensen +1 · 2 citations
Computer Science · Mathematics · #Algebra over a field #Cryptography and Residue Arithmetic #Mathematics #Numerical Methods and Algorithms #Polynomial and algebraic computation #Pure mathematics #math.AC #math.CO #msc:13P10
paper · pdf · doi:10.1090/s0025-5718-07-01986-2
published as Math. Comp. 76 (2007), 2189-2212 · 26 pages
arxiv created 2005/09/23 · openalex publication_date 2007/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper presents algorithms for computing the Gröbner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are based on a uniform definition of the Gröbner 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 Gröbner 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 Gröbner 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.