The CaTS Homepage

Software package written by Anders Nedergaard Jensen
anders at soopadoopa dk

(for version 2.2)

CaTS is a software package whose main functions enumerate (1) all reduced Groebner bases of a lattice ideal, and (2) all monomial A-graded ideals that are flip-connected to a given one. Several variants of these enumeration algorithms are supported such as restricting the above enumerations to all initial ideals or monomial A-graded ideals with a fixed radical. CaTS supports several additional commands among which the highlights are:
  • cats_interactive which allows interactive walks on the edge graph of the state polytope of a toric ideal,
  • cats_fiber which enumerates all monomials of a fixed A-degree, and
  • cats_monomialideal2standardpairs which computes the standard pair decomposition of a monomial ideal.
The full list of commands can be found in appendix A.

CaTS began as a re-implementation of TiGERS, a software package to compute state polytopes of toric ideals, written by Birkett Huber based on algorithms in [Huber & Thomas]. The first version was developed for Jensen's Masters thesis at the University of Aarhus in Denmark. The program has been expanded since then. CaTS is faster than TiGERS (by a factor of 30 on some examples [Jensen]) and has several additional functionalities. To access the full range of commands, CaTS needs to be linked to the packages TOPCOM [Rambau], 4ti2 [Hemmecke & Hemmecke], Normaliz [Bruns & Koch], Macaulay 2 [Grayson & Stillman}, SoPlex [Wunderling] and cdd [Fukuda]. The last two packages are used to solve linear programs and have considerably increased the numerical stability of CaTS. The basic installation only requires linking to the LP solvers.

You can view the manual as a pdf-file or a postscript-file.



CaTS can compute the the 206444 reduced Groebner bases of the ideal associated to the matrix A9 in 529 seconds on a 1333MHz AMD Athlon processor. A9 is the matrix with a single row: (1,2,3,4,5,6,7,8,9).
For more performance results view the table performance.html .

Citations, research and examples

      author = {Jensen, Anders N.},
      title = {CaTS, a software system for toric state polytopes},
      howpublished = {Available at \href{}}
If you use CaTS for your research kindly send me a link to the work. If you compute interesting/large examples using CaTS I will be glad to list it on this page.