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D.4.9.4 primeClosure

Procedure from library normal.lib (see normal_lib).

Usage:
primeClosure(L [,c]); L a list of a ring containing a prime ideal ker, c an optional integer

Return:
a list L consisting of rings L[1],...,L[n] such that
- L[1] is a copy of (not a reference to!) the input ring L[1] - all rings L[i] contain ideals ker, L[2],...,L[n] contain ideals phi such that
L[1]/ker --> ... --> L[n]/ker
are injections given by the corresponding ideals phi, and L[n]/ker is the integral closure of L[1]/ker in its quotient field. - all rings L[i] contain a polynomial nzd such that elements of L[i]/ker are quotients of elements of L[i-1]/ker with denominator nzd via the injection phi.

Note:
- L is constructed by recursive calls of primeClosure itself. - c determines the choice of nzd:
- c not given or equal to 0: first generator of the ideal SL, the singular locus of Spec(L[i]/ker)
- c<>0: the generator of SL with least number of monomials.

Example:
 
LIB "normal.lib";
ring R=0,(x,y),dp;
ideal I=x4,y4;
def K=ReesAlgebra(I)[1];        // K contains ker such that K/ker=R[It]
list L=primeClosure(K);
def R(1)=L[1];                  // L[4] contains ker, L[4]/ker is the
def R(4)=L[4];                  // integral closure of L[1]/ker
setring R(1);
R(1);
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering dp
==> //                  : names    x y U(1) U(2) 
==> //        block   2 : ordering C
ker;
==> ker[1]=y^4*U(1)-x^4*U(2)
setring R(4);
R(4);
==> //   characteristic : 0
==> //   number of vars : 7
==> //        block   1 : ordering a
==> //                  : names    T(1) T(2) T(3) T(4) T(5) T(6) T(7) 
==> //                  : weights     1    1    1    1    1    1    1 
==> //        block   2 : ordering dp
==> //                  : names    T(1) T(2) T(3) T(4) T(5) T(6) T(7) 
==> //        block   3 : ordering C
ker;
==> ker[1]=T(2)*T(5)-T(1)*T(7)
==> ker[2]=T(1)*T(5)-T(2)*T(6)
==> ker[3]=T(5)*T(6)-T(3)*T(7)
==> ker[4]=T(4)*T(6)-T(5)*T(7)
==> ker[5]=T(5)^2-T(6)*T(7)
==> ker[6]=T(4)*T(5)-T(7)^2
==> ker[7]=T(3)*T(5)-T(6)^2
==> ker[8]=T(2)^2*T(6)-T(1)^2*T(7)
==> ker[9]=T(3)*T(4)-T(6)*T(7)
==> ker[10]=T(1)*T(4)-T(2)*T(7)
==> ker[11]=T(2)*T(3)-T(1)*T(6)
==> ker[12]=T(2)^2*T(6)^2-T(1)^2*T(6)*T(7)


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