Singular

Hi all,

a) I am looking for reference(s) elaborating on the factorizing Groebner basis algorithm.
I want to know what it exactly does and how to possibly have a look at the code to see how it is implemented.

b) I had a look at https://www.singular.uni-kl.de/Manual/4-1-3/sing_242.htm#SEC282.
Do I understand it correctly that if facstd(ideal_1, ideal_2) is called with two arguments, then the union of the solutions to the output ideals contains [b]all[/b] solutions of ideal_1 for which all members of ideal_2! are strictly non-zero?!

Let's have a look at the example provided in the link above:
[code] ring r=32003,(x,y,z),(c,dp);
ideal I=xyz,x2z;
facstd(I);
==> [1]:
==> _[1]=z
==> [2]:
==> _[1]=x[/code]

So far,so good. I can see that the variety of I is the union of the x-y-plane and the y-z-plane.
But now:
[code]facstd(I,x);
==> [1]:
==> _[1]=z[/code]
The answer is the x-y-plane which obviously contains (x=0, y, z=0). So it includes the whole y-axis for which x=0!
So am I right or wrong or am I?!