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D.4.25.4 toricRingFromBinomials
Procedure from library normaliz.lib (see normaliz_lib).
- Usage:
- toricRingFromBinomials(ideal I);
toricRingFromBinomials(ideal I, intvec grading);
- Return:
- The ideal 253#253 is generated by binomials of type 1019#1019 (multiindex notation)
in the surrounding polynomial ring
1020#1020. The binomials
represent a congruence on the monoid 1021#1021 with residue monoid 13#13.
Let 420#420 be the image of 13#13 in gp(13#13)/torsion. Then 420#420 is universal in the
sense that every homomorphism from 13#13 to an affine monoid factors through 420#420.
If 253#253 is a prime ideal, then 1022#1022. In general, 1023#1023 where
1024#1024 is the unique minimal prime ideal of 253#253 generated by binomials of type
1019#1019.
The function computes 1025#1025 and returns a newly created
polynomial ring of the same Krull dimension, whose variables are
1026#1026, where 301#301 is the rank of the matrix with rows 1027#1027.
(In general there is no canonical choice for such an embedding.)
The function returns the input ideal I if an option blocking
the computation of Hilbert bases has been activated.
However, in this case some numerical invariants are computed, and
some other data may be contained in files that you can read into
Singular (see showNuminvs, exportNuminvs).
Example:
| LIB "normaliz.lib";
ring R = 37,(u,v,w,x,y,z),dp;
ideal I = u2v-xyz, ux2-wyz, uvw-y2z;
def S = toricRingFromBinomials(I);
setring S;
I;
==> I[1]=x(3)
==> I[2]=x(1)
==> I[3]=x(2)*x(3)^3
==> I[4]=x(1)*x(2)*x(3)^2
==> I[5]=x(1)^3*x(2)
==> I[6]=x(1)^2*x(2)^3*x(3)^5
| See also:
normalToricRingFromBinomials;
toricRingFromBinomials.
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