These are wrappers of some use cases of 4ti2s groebner command.

`‣ 4ti2Interface_groebner_matrix` ( matrix[, ordering] ) | ( function ) |

Returns: 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

`‣ 4ti2Interface_groebner_basis` ( basis[, ordering] ) | ( function ) |

Returns: 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.

We want to compute the groebner basis of the ideal defining the affine toric variety associated to the cone generated by the inequalities [ [ 7, -1 ], [ 0, 1 ] ], i.e. a rational normal curve.

gap> cone := [ [ 7, -1 ], [ 0, 1 ] ]; [ [ 7, -1 ], [ 0, 1 ] ] gap> basis := 4ti2Interface_hilbert_inequalities( cone );; gap> Sort( basis ); gap> basis; [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ] ] gap> groebner := 4ti2Interface_groebner_matrix( basis );; gap> Sort( groebner ); gap> groebner; [ [ -1, 0, 0, 1, 1, 0, 0, -1 ], [ -1, 0, 0, 2, 0, 0, -1, 0 ], [ -1, 0, 1, 0, 0, 1, 0, -1 ], [ -1, 0, 1, 0, 1, 0, -1, 0 ], [ -1, 0, 1, 1, 0, -1, 0, 0 ], [ -1, 0, 2, 0, -1, 0, 0, 0 ], [ -1, 1, 0, 0, 0, 0, 1, -1 ], [ -1, 1, 0, 0, 0, 1, -1, 0 ], [ -1, 1, 0, 0, 1, -1, 0, 0 ], [ -1, 1, 0, 1, -1, 0, 0, 0 ], [ -1, 1, 1, -1, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 2, 0, 0, -1 ], [ 0, -1, 0, 1, 0, 1, 0, -1 ], [ 0, -1, 1, 0, 0, 0, 1, -1 ], [ 0, 0, -1, 0, 1, 1, 0, -1 ], [ 0, 0, -1, 1, 0, 0, 1, -1 ], [ 0, 0, 0, -1, 0, 2, 0, -1 ], [ 0, 0, 0, -1, 1, 0, 1, -1 ], [ 0, 0, 0, 0, -1, 1, 1, -1 ], [ 0, 0, 0, 0, 0, -1, 2, -1 ] ] gap> Length( groebner ); 21

These are wrappers of some use cases of 4ti2s hilbert command.

`‣ 4ti2Interface_hilbert_inequalities` ( A ) | ( function ) |

`‣ 4ti2Interface_hilbert_inequalities_in_positive_orthant` ( A ) | ( function ) |

Returns: a list of vectors

This function produces the hilbert basis of the cone C given by `A`x >= 0 for all x in C. For the second function also x >= 0 is assumed.

`‣ 4ti2Interface_hilbert_equalities_in_positive_orthant` ( A ) | ( function ) |

Returns: a list of vectors

This function produces the hilbert basis of the cone C given by the equations `A`x = 0 in the positive orthant of the coordinate system.

`‣ 4ti2Interface_hilbert_equalities_and_inequalities` ( A, B ) | ( function ) |

`‣ 4ti2Interface_hilbert_equalities_and_inequalities_in_positive_orthant` ( A, B ) | ( function ) |

Returns: a list of vectors

This function produces the hilbert basis of the cone C given by the equations `A`x = 0 and the inequations `B`x >= 0. For the second function x>=0 is assumed.

We want to compute the Hilbert basis of the cone obtained by intersecting the positive orthant with the hyperplane given by the equation below.

gap> gens := [ 23, 25, 37, 49 ]; [ 23, 25, 37, 49 ] gap> equation := [ Concatenation( gens, -gens ) ]; [ [ 23, 25, 37, 49, -23, -25, -37, -49 ] ] gap> basis := 4ti2Interface_hilbert_equalities_in_positive_orthant( equation );; gap> Length( basis ); 436

We want to compute the Hilbert basis of the cone which faces are represented by the inequalities below. This example is taken from the toric and the ToricVarieties package manual. In both packages it is very slow with the internal algorithms.

gap> inequalities := [ [1,2,3,4], [0,1,0,7], [3,1,0,2], [0,0,1,0] ]; [ [ 1, 2, 3, 4 ], [ 0, 1, 0, 7 ], [ 3, 1, 0, 2 ], [ 0, 0, 1, 0 ] ] gap> basis := 4ti2Interface_hilbert_inequalities( inequalities );; gap> Length( basis ); 29

`‣ 4ti2Interface_zsolve_equalities_and_inequalities` ( eqs, eqs_rhs, ineqs, ineqs_rhs[, signs] ) | ( function ) |

`‣ 4ti2Interface_zsolve_equalities_and_inequalities_in_positive_orthant` ( eqs, eqs_rhs, ineqs, ineqs_rhs ) | ( function ) |

Returns: 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.

`‣ 4ti2Interface_graver_equalities` ( eqs[, signs] ) | ( function ) |

`‣ 4ti2Interface_graver_equalities_in_positive_orthant` ( eqs ) | ( function ) |

Returns: 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 \(x_i \geq 0\).

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