4ti2 functions
Groebner These are wrappers of some use cases of 4ti2s groebner command. A list of vectors This launches the 4ti2 groebner command with the argument as matrix input. The output will be the the Groebner basis of the binomial ideal generated by the left kernel of the input matrix. Note that this is different from 4ti2's convention which takes the right kernel. It returns the output of the groebner command as a list of lists. The second argument can be a vector to specify a monomial ordering, in the way that x^m > x^n if ordering*m > ordering*n A list of vectors This launches the 4ti2 groebner command with the argument as matrix input. The outpur will be the Groebner basis of the binomial ideal generated by the rows of the input matrix. It returns the output of the groebner command as a list of lists. The second argument is like before. <#Include Label="Groebner1">
Hilbert These are wrappers of some use cases of 4ti2s hilbert command. a list of vectors This function produces the hilbert basis of the cone C given by Ax >= 0 for all x in C. For the second function also x >= 0 is assumed.

a list of vectors This function produces the hilbert basis of the cone C given by the equations Ax = 0 in the positive orthant of the coordinate system. a list of vectors This function produces the hilbert basis of the cone C given by the equations Ax = 0 and the inequations Bx >= 0. For the second function x>=0 is assumed.

<#Include Label="HilbertBasis"> <#Include Label="HilbertBasis2">
ZSolve a list of three matrices This function produces a basis of the system eqs = eqs_rhs and ineqs >= ineqs_rhs. It outputs a list containing three matrices. The first one is a list of points in a polytope, the second is the hilbert basis of a cone. The set of solutions is then the minkowski sum of the polytope generated by the points in the first list and the cone generated by the hilbert basis in the second matrix. The third one is the free part of the solution polyhedron. The optional argument signs must be a list of zeros and ones which length is the number of variables. If the ith entry is one, the ith variable must be >= 0. If the entry is 0, the number is arbitraty. Default is all zero. It is also possible to set the option precision to 32, 64 or gmp. The default, if no option is given, 32 is used. Please note that a higher precision leads to slower computation. For the second function xi >= 0 for all variables is assumed.
Graver a matrix This calls the function graver with the equalities eqs = 0. It outputs one list containing the graver basis of the system. the optional argument signs is used like in zsolve. The second command assumes x_i \geq 0.