#### D.4.36.1 toric_ideal

Procedure from library `toric.lib` (see toric_lib).

Usage:
  toric_ideal(A,alg); A intmat, alg string
  toric_ideal(A,alg,prsv); A intmat, alg string, prsv intvec

Return:
  ideal: standard basis of the toric ideal of A

Note:
  These procedures return the standard basis of the toric ideal of A with respect to the term ordering in the current basering. Not all term orderings are supported: The usual global term orderings may be used, but no block orderings combining them.
  One may call the procedure with several different algorithms:
  - the algorithm of Conti/Traverso using elimination (ect),
  - the algorithm of Pottier (pt),
  - an algorithm of Bigatti/La Scala/Robbiano (blr),
  - the algorithm of Hosten/Sturmfels (hs),
  - the algorithm of DiBiase/Urbanke (du).
  The argument `alg' should be the abbreviation for an algorithm as above: ect, pt, blr, hs or du.

  If `alg' is chosen to be `blr' or `hs', the algorithm needs a vector with positive coefficients in the row space of A.
  If no row of A contains only positive entries, one has to use the second version of toric_ideal which takes such a vector as its third argument.
  For the mathematical background, see

  Toric ideals and integer programming.

Example:

    LIB "toric.lib";
    ring r=0,(x,y,z),dp;
    // call with two arguments
    intmat A[2][3]=1,1,0,0,1,1;
    A;
    ==> 1,1,0,
    ==> 0,1,1
    ideal I=toric_ideal(A,"du");
    I;
    ==> I[1]=xz-y
    I=toric_ideal(A,"blr");
    ==>    ? The chosen algorithm needs a positive vector in the row space of the\
        matrix.
    ==>    ? leaving toric.lib::toric_ideal_1 (0)
    ==>    ? leaving toric.lib::toric_ideal (704)
    I;
    ==> I[1]=xz-y
    // call with three arguments
    intvec prsv=1,2,1;
    I=toric_ideal(A,"blr",prsv);
    I;
    ==> I[1]=xz-y

See also: Toric ideals; intprog_lib; toric_std.

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