A zero-dimensional polynomial ideal may have a lot of complex zeros. But
sometimes, only some of them are needed. In this paper, for a zero-dimensional
ideal I, we study its complex zeros that locate in another variety
V(J) where J is an arbitrary ideal.
The main problem is that for a point in V(I)∩V(J)=V(I+J), its multiplicities w.r.t. I and I+J may be
different. Therefore, we cannot get the multiplicity of this point w.r.t. I
by studying I+J. A straightforward way is that first compute the points of
V(I+J), then study their multiplicities w.r.t. I. But the former
step is difficult to realize exactly.
In this paper, we propose a natural geometric explanation of the localization
of a polynomial ring corresponding to a semigroup order. Then, based on this
view, using the standard basis method and the border basis method, we introduce
a way to compute the complex zeros of I in V(J) with their
multiplicities w.r.t. I. As an application, we compute the sum of Milnor
numbers of the singular points on a polynomial hypersurface and work out all
the singular points on the hypersurface with their Milnor numbers