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

TL;DR

This paper presents a geometric approach to compute specific complex zeros of zero-dimensional polynomial ideals within a given variety, along with their multiplicities, using standard and border basis methods.

Contribution

It introduces a geometric explanation of localization via semigroup orders and develops methods to compute zeros and multiplicities in constrained varieties.

Findings

Computed zeros of polynomial ideals in specified varieties.

Determined multiplicities of zeros with respect to the ideal.

Applied methods to find singular points and Milnor numbers on hypersurfaces.

Abstract

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 $\textbf{V}(J)$ where $J$ is an arbitrary ideal. The main problem is that for a point in $\textbf{V}(I) \cap \textbf{V}(J)=\textbf{V}(I+J)$, 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 $\textbf{V}(I + J)$, 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 $\textbf{V}(J)$ 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.