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

##### MPS-Authors
/persons/resource/persons101876

Liang,  Ye
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Fulltext (public)

arXiv:1408.3639.pdf
(Preprint), 174KB

##### Supplementary Material (public)
There is no public supplementary material available
##### Citation

Liang, Y. (2014). Solving Polynomial Equations with Equation Constraints: the Zero-dimensional Case. Retrieved from http://arxiv.org/abs/1408.3639.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0024-5B43-5
##### 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.