<#GAPDoc Label="Groebner1">
 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.
<Example><![CDATA[
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
]]></Example>


<#/GAPDoc>
<#GAPDoc Label="HilbertBasis">
 We want to compute the Hilbert basis of the cone obtained
 by intersecting the positive orthant with the hyperplane
 given by the equation below.
<Example><![CDATA[
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
]]></Example>


<#/GAPDoc>
<#GAPDoc Label="HilbertBasis2">
 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.
<Example><![CDATA[
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
]]></Example>


<#/GAPDoc>
