 ring r=0,(x,y,z),ds;
ideal i=x^4y*z^2,x*yz^3,y^2x^3*z; // the space curve singularity
qhweight(i);
==> 1,2,1
// The given space curve singularity is quasihomogeneous. Hence we can pass
// to the polynomial ring.
ring rr=0,(x,y,z),dp;
ideal i=imap(r,i);
resolution ires=mres(i,0);
ires;
==> 1 3 2
==> rr < rr < rr
==>
==> 0 1 2
==>
// From the structure of the resolution, we see that the CohenMacaulay
// type of the given singularity is 2
//
// Let us now look for the branches using the primdec library.
LIB "primdec.lib";
primdecSY(i);
==> [1]:
==> [1]:
==> _[1]=z3xy
==> _[2]=x3+x2z+xz2+xy+yz
==> _[3]=x2z2+x2y+xyz+yz2+y2
==> [2]:
==> _[1]=z3xy
==> _[2]=x3+x2z+xz2+xy+yz
==> _[3]=x2z2+x2y+xyz+yz2+y2
==> [2]:
==> [1]:
==> _[1]=xz
==> _[2]=z2y
==> [2]:
==> _[1]=xz
==> _[2]=z2y
def li=_[1];
ideal i2=li[2]; // call the first ideal i1
// The curve seems to have 2 branches by what we computed using the
// algorithm of ShimoyamaYokoyama.
// Now the same computation by the GianniTragerZacharias algorithm:
primdecGTZ(i);
==> [1]:
==> [1]:
==> _[1]=z2+y
==> _[2]=xz
==> [2]:
==> _[1]=z2+y
==> _[2]=xz
==> [2]:
==> [1]:
==> _[1]=z8+yz6+y2z4+y3z2+y4
==> _[2]=xz5+z6+yz4+y2z2+y3
==> _[3]=z3+xy
==> _[4]=x2z2+xz3+xyz+yz2+y2
==> _[5]=x3+x2z+xz2+xy+yz
==> [2]:
==> _[1]=z8+yz6+y2z4+y3z2+y4
==> _[2]=xz5+z6+yz4+y2z2+y3
==> _[3]=z3+xy
==> _[4]=x2z2+xz3+xyz+yz2+y2
==> _[5]=x3+x2z+xz2+xy+yz
// Having computed the primary decomposition in 2 different ways and
// having obtained the same number of branches, we might expect that the
// number of branches is really 2, but we can check this by formulae
// for the invariants of space curve singularities:
//
// mu = tau  t + 1 (for quasihomogeneous curve singularities)
// where mu denotes the Milnor number, tau the Tjurina number and
// t the CohenMacaulay type
//
// mu = 2 delta  r + 1
// where delta denotes the deltaInvariant and r the number of branches
//
// tau can be computed by using the corresponding procedure T1 from
// sing.lib.
setring r;
LIB "sing.lib";
T_1(i);
==> // dim T_1 = 13
==> _[1]=gen(6)+2z*gen(5)
==> _[2]=gen(4)+3x2*gen(2)
==> _[3]=gen(3)+gen(1)
==> _[4]=x*gen(5)y*gen(2)z*gen(1)
==> _[5]=x*gen(1)z2*gen(2)
==> _[6]=y*gen(5)+3x2z*gen(2)
==> _[7]=y*gen(2)z*gen(1)
==> _[8]=2y*gen(1)z2*gen(5)
==> _[9]=z2*gen(5)
==> _[10]=z2*gen(1)
==> _[11]=x3*gen(2)
==> _[12]=x2z2*gen(2)
==> _[13]=xz3*gen(2)
==> _[14]=z4*gen(2)
setring rr;
// Hence tau is 13 and therefore mu is 12. But then it is impossible that
// the singularity has two branches, since mu is even and delta is an
// integer!
// So obviously, we did not decompose completely. Because the second branch
// is smooth, only the first ideal can be the one which can be decomposed
// further.
// Let us now consider the normalization of this first ideal i1.
LIB "normal.lib";
normal(i2);
==>
==> // 'normal' created a list, say nor, of two elements.
==> // To see the list type
==> nor;
==>
==> // * nor[1] is a list of 1 ring(s).
==> // To access the ith ring nor[1][i], give it a name, say Ri, and type
==> def R1 = nor[1][1]; setring R1; norid; normap;
==> // For the other rings type first (if R is the name of your base ring)
==> setring R;
==> // and then continue as for R1.
==> // Ri/norid is the affine algebra of the normalization of R/P_i where
==> // P_i is the ith component of a decomposition of the input ideal id
==> // and normap the normalization map from R to Ri/norid.
==>
==> // * nor[2] is a list of 1 ideal(s). Let ci be the last generator
==> // of the ideal nor[2][i]. Then the integral closure of R/P_i is
==> // generated as Rsubmodule of the total ring of fractions by
==> // 1/ci * nor[2][i].
==> [1]:
==> [1]:
==> // coefficients: QQ
==> // number of vars : 6
==> // block 1 : ordering dp
==> // : names T(1) T(2) T(3)
==> // block 2 : ordering dp
==> // : names x y z
==> // block 3 : ordering C
==> [2]:
==> [1]:
==> _[1]=y
==> _[2]=xz
==> _[3]=x2
==> _[4]=z2
def rno=_[1][1];
setring rno;
norid;
==> norid[1]=T(2)*z+x
==> norid[2]=T(1)*xz
==> norid[3]=T(2)*xT(3)*z
==> norid[4]=T(1)*z+T(2)*z+T(3)*x+T(3)*z+z
==> norid[5]=T(2)*y+z^2
==> norid[6]=T(1)*z^2y
==> norid[7]=T(2)*z^2T(3)*y
==> norid[8]=T(1)*y+T(2)*y+T(3)*z^2+T(3)*y+y
==> norid[9]=T(1)^2+T(1)+T(2)+T(3)+1
==> norid[10]=T(1)*T(2)1
==> norid[11]=T(2)^2T(3)
==> norid[12]=T(1)*T(3)T(2)
==> norid[13]=T(2)*T(3)+T(1)+T(2)+T(3)+1
==> norid[14]=T(3)^2T(1)
==> norid[15]=z^3x*y
==> norid[16]=x^3+x^2*z+x*z^2+x*y+y*z
==> norid[17]=x^2*z^2+x^2*y+x*y*z+y*z^2+y^2
// The ideal is generated by a polynomial in one variable of degree 4 which
// factors completely into 4 polynomials of type T(2)+a.
// From this, we know that the ring of the normalization is the direct sum of
// 4 polynomial rings in one variable.
// Hence our original curve has these 4 branches plus a smooth one
// which we already determined by primary decomposition.
// Our final result is therefore: 5 branches.
