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. 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.

References in zbMATH (referenced in 105 articles , 1 standard article )

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  1. Morrison, Ian; Swinarski, David: Gröbner techniques for low-degree Hilbert stability (2011)
  2. Cueto, Maria Angélica; Morton, Jason; Sturmfels, Bernd: Geometry of the restricted Boltzmann machine (2010)
  3. Cueto, María Angélica; Tobis, Enrique A.; Yu, Josephine: An implicitization challenge for binary factor analysis (2010)
  4. De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco: Triangulations. Structures for algorithms and applications (2010)
  5. Eröcal, Burçin; Stein, William: The Sage project: unifying free mathematical software to create a viable alternative to Magma, Maple, Mathematica and Matlab (2010)
  6. Hampton, Marshall: Finiteness of kite relative equilibria in the five-vortex and five-body problems (2010)
  7. Jensen, Anders Nedergaard: Traversing symmetric polyhedral fans (2010)
  8. Markwig, Thomas: Computer algebra methods in tropical geometry (2010)
  9. Herrmann, Sven: Splits and tight spans of convex polytopes (2009)
  10. Morita, Tomohito: Hilbert schemes of finite abelian group orbits and Gröbner fans (2009)
  11. Vasiliev, N. N.; Pavlov, D. A.: Enumeration of finite monomial orderings and combinatorics of universal Gröbner bases (2009)
  12. Verschelde, Jan: Polyhedral methods in numerical algebraic geometry (2009)
  13. Jensen, Anders Nedergaard: Computing Gröbner fans and tropical varieties in \textttGfan (2008)
  14. Jensen, Anders Nedergaard; Markwig, Hannah; Markwig, Thomas: An algorithm for lifting points in a tropical variety (2008)
  15. Stillman, Michael E. (ed.); Takayama, Nobuki (ed.); Verschelde, Jan (ed.): Software for algebraic geometry. Papers of a workshop, Minneapolis, MN, USA, October 23--27, 2006 (2008)
  16. Bahloul, Rouchdi; Takayama, Nobuki: Local Gröbner fans (2007)
  17. Bogart, T.; Jensen, A. N.; Speyer, D.; Sturmfels, B.; Thomas, R. R.: Computing tropical varieties (2007)
  18. Bogart, Tristram; Jensen, Anders N.; Thomas, Rekha R.: The circuit ideal of a vector configuration (2007)
  19. Craw, Alastair; Maclagan, Diane; Thomas, Rekha R.: Moduli of McKay quiver representations. II: Gröbner basis techniques (2007)
  20. Dickenstein, Alicia; Feichtner, Eva Maria; Sturmfels, Bernd: Tropical discriminants (2007)