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D.4.16.3 normalToricRingFromBinomials
Procedure from library normaliz.lib (see normaliz_lib).
- Usage:
- normalToricRingFromBinomials(ideal I);
- Return:
- @tex
The ideal $I$ is generated by binomials of type $X^a-X^b$ (multiindex notation)
in the surrounding polynomial ring $K[X]=K[X_1,...,X_n]$. The binomials
represent a congruence on the monoid ${Z}^n$ with residue monoid $M$.
Let $N$ be the image of $M$ in gp($M$)/torsion. Then $N$ is universal in the
sense that every homomorphism from $M$ to an affine monoid factors through $N$.
If $I$ is a prime ideal, then $K[N]= K[X]/I$. In general, $K[N]=K[X]/P$ where
$P$ is the unique minimal prime ideal of $I$ generated by binomials of type
$X^a-X^b$.
The function computes the normalization of $K[N]$ and returns a newly created
polynomial ring of the same Krull dimension, whose variables are
$x(1),...,x(n-r)$, where $r$ is the rank of the matrix with rows $a-b$.
(In general there is no canonical choice for such an embedding.)
Inside this polynomial ring there is an ideal $I$ which lists the algebra
generators of the normalization of $K[N]$.
@end tex
The function returns the input ideal I if one of the options
supp , triang , or hvect 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.
Example:
| LIB "normaliz.lib";
ring R = 37,(u,v,w,x,y,z),dp;
ideal I = u2v-xyz, ux2-wyz, uvw-y2z;
def S = normalToricRingFromBinomials(I);
setring S;
I;
==> I[1]=x(1)
==> I[2]=x(3)
==> I[3]=x(1)^2*x(2)
==> I[4]=x(1)*x(2)*x(3)
==> I[5]=x(2)*x(3)^2
==> I[6]=x(1)^3*x(2)^2
==> I[7]=x(1)^2*x(2)^2*x(3)
==> I[8]=x(1)*x(2)^2*x(3)^2
==> I[9]=x(1)*x(2)^3*x(3)^3
| See also:
ehrhartRing;
intclMonIdeal;
intclToricRing;
normalToricRing.
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