Solving Polynomial Equations with Equation Constraints: the Zero-dimensional Case

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.