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D.4.5.3 curveDeligneNumber

Procedure from library curveInv.lib (see curveInv_lib).

Usage:
curveDeligneNumber(I); I ideal

Assume:
I is a radical ideal, dim(R/I) = 1

Return:
the Deligne number of R/I

Remarks:
The Deligne number e satisfies by definition: e = 3delta - m. So the algorithm splits the computation into two parts: one part computes the delta invariant, the other part the colength of derivations m.

Note:
the optional parameter can be used if the normalization has already been computed. If a list L contains the output of the procedure normal (with options prim, wd and usering if the ring has a mixed ordering), apply curveDeligneNumber(I,L)

Example:
 
LIB "curveInv.lib";
//////////////////////////////
// Deligne number of curves //
//////////////////////////////
// Example 1:
ring R = 0,(x,y,z),ds;
ideal I = x2-y4z,z3y2+xy2;
I = std(radical(I));
curveDeligneNumber(I);
==> 30
// Example 2:
ring S = 0,(x,y),ds;
ideal I = (x+y)*(x2-y3);
curveDeligneNumber(I);
==> 5
// Example 3:
ideal J = (x2-y3)*(x2+y2)*(x-y);
curveDeligneNumber(J);
==> 15
// Let us also compute the milnor number of this complete intersection:
milnor(J);
==> 17
// We see that the Milnor number is bigger than the Deligne number. Hence, this
// curve cannot be quasi homogeneous. This can also be verified by Saitos criterion:
reduce(J[1],std(jacob(J[1])));
==> -1/5y6+19/50y7
See also: curveColengthDerivations; curveDeltaInv.