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 121 articles , 1 standard article )

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  1. Dimitrova, Elena; Hu, Jingzhen; Liang, Qingzhong; Stigler, Brandilyn; Zhang, Anyu: Algebraic model selection and experimental design in biological data science (2022)
  2. Panizzut, Marta; Vigeland, Magnus Dehli: Tropical lines on cubic surfaces (2022)
  3. Balletti, Gabriele; Panizzut, Marta; Sturmfels, Bernd: (K3) polytopes and their quartic surfaces (2021)
  4. Brandt, Madeline; Eur, Christopher; Zhang, Leon: Tropical flag varieties (2021)
  5. Corey, Daniel: Initial degenerations of Grassmannians (2021)
  6. Görlach, Paul; Ren, Yue; Sommars, Jeff: Detecting tropical defects of polynomial equations (2021)
  7. Vaccon, Tristan; Verron, Thibaut; Yokoyama, Kazuhiro: On affine tropical F5 algorithms (2021)
  8. Donten-Bury, Maria; Görlach, Paul; Wrobel, Milena: Towards classifying toric degenerations of cubic surfaces (2020)
  9. Markwig, Thomas; Ren, Yue: Computing tropical varieties over fields with valuation (2020)
  10. Morrison, Ralph: Tropical geometry (2020)
  11. Perry, John: A dynamic F4 algorithm to compute Gröbner bases (2020)
  12. Rossi, Michele; Terracini, Lea: Toric varieties and Gröbner bases: the complete (\mathbbQ)-factorial case (2020)
  13. Tillmann, Stephan: Tropical varieties associated to ideal triangulations: the Whitehead link complement (2020)
  14. Chan, Andrew J.; Maclagan, Diane: Gröbner bases over fields with valuations (2019)
  15. Chen, Tianran: Unmixing the mixed volume computation (2019)
  16. Cueto, Maria Angelica; Markwig, Hannah: Tropical geometry of genus two curves (2019)
  17. Jensen, Anders; Ren, Yue; Schönemann, Hans: The gfanlib interface in Singular and its applications (2019)
  18. Leykin, Anton; Yu, Josephine: Beyond polyhedral homotopies (2019)
  19. Ren, Yue; Martini, Johannes W. R.; Torres, Jacinta: Decoupled molecules with binding polynomials of bidegree ((n,2)) (2019)
  20. Tripoli, Paolo: Tropical Chow hypersurfaces (2019)

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