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D.4.6.1 normal
Procedure from library normal.lib (see normal_lib).
- Usage:
normal(i [,choose]); i a radical ideal, choose empty, 1 or "wd" if choose=1 the normalization of the associated primes is computed (which is sometimes more efficient); if
choose="wd"the delta invariant is computed simultaneously; this may take much more time in the reducible case, since the factorizing standard basis algorithm cannot be used.- Assume:
The ideal must be radical, for non-radical ideals the output may be wrong (i=radical(i); makes i radical)
- Return:
a list of rings, say nor and in case of
choose="wd"an integer at the end of the list. Each ringnor[i]contains two ideals with given namesnoridandnormapsuch that
- the direct sum of the ringsnor[i]/noridis the normalization of basering/id;
-normapgives the normalization map from basering/id tonor[i]/norid(for each i).- Note:
to use the i-th ring type:
def R=nor[i]; setring R;.
Increasing printlevel displays more comments (default: printlevel=0).
Not implemented for local or mixed orderings.
If the input ideal i is weighted homogeneous a weighted ordering may be used (qhweight(i); computes weights).
Example:
LIB "normal.lib"; ring r=32003,(x,y,z),wp(2,1,2); ideal i=z3-xy4; list nor=normal(i); → → // 'normal' created a list of 1 ring(s). → // nor[1+1] is the delta-invariant in case of choose=wd. → // To see the rings, type (if the name of your list is nor): → show( nor); → // To access the 1-st ring and map (similar for the others), type: → def R = nor[1]; setring R; norid; normap; → // R/norid is the 1-st ring of the normalization and → // normap the map from the original basering to R/norid show(nor); → // list, 1 element(s): → [1]: → // ring: (32003),(T(1),T(2),T(3)),(a(2,1,1),dp(3),C); → // minpoly = 0 → // objects belonging to this ring: → // normap [0] ideal, 3 generator(s) → // norid [0] ideal, 1 generator(s) def r1=nor[1]; setring r1; norid; → norid[1]=T(3)3-T(1)T(2) normap; → normap[1]=T(1) → normap[2]=T(2) → normap[3]=T(2)T(3) ring s=0,(x,y),dp; ideal i=(x-y^2)^2 - y*x^3; nor=normal(i,"wd"); → → // 'normal' created a list of 1 ring(s). → // nor[1+1] is the delta-invariant in case of choose=wd. → // To see the rings, type (if the name of your list is nor): → show( nor); → // To access the 1-st ring and map (similar for the others), type: → def R = nor[1]; setring R; norid; normap; → // R/norid is the 1-st ring of the normalization and → // normap the map from the original basering to R/norid //the delta-invariant nor[size(nor)]; → 3 |