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D.4.11.2 toric_std
Procedure from library toric.lib (see toric_lib).
- Usage:
toric_std(I); I ideal
- Return:
ideal: standard basis of I
- Note:
This procedure computes the standard basis of I using a specialized Buchberger algorithm. The generating system by which I is given has to consist of binomials of the form x^u-x^v. There is no real check if I is toric. If I is generated by binomials of the above form, but not toric, toric_std computes an ideal ‘between’ I and its saturation with respect to all variables.
For the mathematical background, see
Example:
LIB "toric.lib"; ring r=0,(x,y,z),wp(3,2,1); // call with toric ideal (of the matrix A=(1,1,1)) ideal I=x-y,x-z; ideal J=toric_std(I); J; → J[1]=y-z → J[2]=x-z // call with the same ideal, but badly chosen generators: // 1) not only binomials I=x-y,2x-y-z; J=toric_std(I); → ERROR: Generator 2 of the input ideal is no difference of monomials. // 2) binomials whose monomials are not relatively prime I=x-y,xy-yz,y-z; J=toric_std(I); → Warning: The monomials of generator 2 of the input ideal are not relative\ ly prime. J; → J[1]=y-z → J[2]=x-z // call with a non-toric ideal that seems to be toric I=x-yz,xy-z; J=toric_std(I); J; → J[1]=y2-1 → J[2]=x-yz // comparison with real standard basis and saturation ideal H=std(I); H; → H[1]=x-yz → H[2]=y2z-z LIB "elim.lib"; sat(H,xyz); → [1]: → _[1]=x-yz → _[2]=y2-1 → [2]: → 1 |
See also: Toric ideals; intprog_lib; toric_ideal; toric_lib.