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D.4.19.3 normalToricRingFromBinomials

Procedure from library normaliz.lib (see normaliz_lib).

normalToricRingFromBinomials(ideal I);
normalToricRingFromBinomials(ideal I, intvec grading);

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 $K[N]$.

The function returns the input ideal I if one of the options supp, triang, volume, or hseries 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).

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[1]=x(3)
==> I[2]=x(2)
==> I[3]=x(1)*x(3)^3
==> I[4]=x(1)*x(2)*x(3)^2
==> I[5]=x(1)*x(2)^2*x(3)
==> I[6]=x(1)*x(2)^3
==> I[7]=x(1)^2*x(2)^3*x(3)^2
==> I[8]=x(1)^2*x(2)^4*x(3)
==> I[9]=x(1)^3*x(2)^5*x(3)^2
See also: ehrhartRing; intclMonIdeal; intclToricRing; normalToricRing.