ring r=0,(x,y,z),ds;
ideal I=x^4-y*z^2,x*y-z^3,y^2-x^3*z;
list L=facstd(I);
L;
==>
[1]:
_[1]=xy-z3
_[2]=y2-x3z
_[3]=yz2-x4
_[4]=x4+x3z+x2z2+xz3+z4
[2]:
_[1]=x-z
_[2]=y-z2
Again we only obtain 2 components. The 4-th generator
of the first ideal shows that this component might contain up
to 4 branches.Note that factorization is incomplete because we are working over the rationals. To see that the first ideal has 4 branches , we move to an algebraic extension:
ideal J=L[1]; // the first ideal
ring s=(0,a),(x,y,z),dp;
minpoly=a4+a3+a2+a+1;
def J=imap(r,J);
LIB "primdec.lib";
J[5]=x-a*z; // a factor of J[4]
minAssGTZ(J); // minimal associated primes
==>
[1]:
_[1]=x+(-a)*z
_[2]=y
_[3]=z
Note also that using the factorizing Gröbner basis algorithm is usually
not sufficient to completely decompose an ideal into its components, it should
be used primarily to obtain smaller problems which are then appropriate for
the normalization algorithm.
<-- Branches of an isolated space curve singularity
<-- computed via Primary Decomposition
<-- computed via Normalization