The Gfan Homepage

Software package written by Anders Nedergaard Jensen


Abstract

Gfan is a software package for computing Gröbner fans and tropical varieties. These are polyhedral fans associated to polynomial ideals. The maximal cones of a Gröbner fan are in bijection with the marked reduced Gröbner bases of its defining ideal. The software computes all marked reduced Gröbner bases of an ideal. Their union is a universal Gröbner basis. The tropical variety of a polynomial ideal is a certain subcomplex of the Gröbner fan. Gfan contains algorithms for computing this complex for general ideals and specialized algorithms for tropical curves, tropical hypersurfaces and tropical varieties of prime ideals. In addition to the above core functions the package contains many tools which are useful in the study of Gröbner bases, initial ideals and tropical geometry. Among these are an interactive traversal program for Gröbner fans and programs for graphical renderings. The full list of commands can be found in Appendix B of the manual. For ordinary Gröbner basis computations Gfan is not competitive in speed compared to programs such as CoCoA, Singular and Macaulay2.


If you do not know what Gfan is you can read a presentation of it in pdf-format or in postscript-format. If you would like to know more you can also view the manual as a pdf-file or a postscript-file. CaTS is another program for computing Gröbner fans working for toric and lattice ideals only.

Download

  • Gfan version 0.3 (October 12 2007, major update, some programs have changed behaviour, a new Polymake compatible format for polyhedral fans and polyhedral cones is supported, free choice of variable names)
  • Gfan version 0.2.2 (July 24 2006, changes in installation script, bug fixes for non-homogeneous ideals in and xfig output, GPL license)
  • Gfan version 0.2.1 (February 5 2006, support for Z/pZ as coefficient field added)
  • Gfan version 0.2 (October 12 2005)
  • Gfan version 0.1 (April 8 2005)

Citations

@Misc{gfan,
     author = {Jensen, Anders N.},
     title = {Gfan, a software system for {G}r{\"o}bner fans},
     howpublished = {Available at \url{http://www.math.tu-berlin.de/~jensen/software/gfan/gfan.html}}
}

References

  • David Avis and Komei Fukuda. A basis enumeration algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Computational Geometry, 8:295--313, 1992.
  • Tristram Bogart, Anders Jensen, David Speyer, Bernd Sturmfels, and Rekha Thomas. Computing tropical varieties. 2005. math.AG/0507563.
  • Stéphane Collart, Michael Kalkbrener, and Daniel Mall. Converting bases with the Gröbner walk. J. Symb. Comput., 24(3/4):465--469, 1997.
  • Komei Fukuda. cddlib reference manual, cddlib Version 093b. Swiss Federal Institute of Technology, Lausanne and Zürich, Switzerland, 2003. http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html .
  • Komei Fukuda, Anders Jensen, and Rekha Thomas. Computing Gröbner fans. 2005. math.AC/0509544.
  • Torbjörn Granlund et al. GNU multiple precision arithmetic library 4.1.2, December 2002. http://swox.com/gmp/.
  • Birkett Huber and Rekha R. Thomas. Computing Gröbner fans of toric ideals. Experimental Mathematics, 9(3/4):321--331, 2000.
  • Anders Jensen. CaTS, a software system for toric state polytopes. Available at http://www.soopadoopa.dk/anders/cats/cats.html.
  • Anders Jensen. A non-regular Gröbner fan. 2005. math.CO/0501352.
  • Teo Mora and Lorenzo Robbiano. The Gröbner fan of an ideal. J. Symb. Comput., 6(2/3):183--208, 1988.
  • Jörg Rambau. Topcom: Triangulations of point configurations and oriented matroids. ZIB report, 02-17, 2002.
  • Bernd Sturmfels. Gröbner bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, 1996.