2014/08/15 by Ye Liang, Liang, Ye
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.SC
paper · pdf · doi:10.48550/arxiv.1408.3639
arxiv created 2014/08/15 · openalex publication_date 2014/08/15 · arxiv updated 2014/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.