[论文解读] How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft II)

Pinaki Mondal|arXiv (Cornell University)|Jun 14, 2018

Polynomial and algebraic computation参考文献 41被引用 3

一句话总结

本文提出了一种基于 торической 几何的方法，用于计算代数闭域上 n 个多项式方程在 n 个变量中的孤立解的个数，将 Bernstein 定理由环面 (k×)ⁿ 扩展至整个仿射空间 kⁿ。主要贡献是通过 Newton 多面体对一般系统进行表征，并利用混合体积给出解数的公式，从而在任意代数闭域上完整证明了解决了代数几何中长期存在的一个难题。

ABSTRACT

In this book we describe an approach through toric geometry to the following problem: estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k. The outcome of this approach is the number of solutions for systems in terms of their polytopes, and an explicit characterization of what makes a system generic. The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic systems on the torus (k\0)^n. In the context of our problem, however, the natural domain of solutions is not the torus, but the affine space k^n. There were a number of works on extension of Bernstein's theorem to the case of affine space, and recently it has been completely resolved, the final steps having been carried out by the author. The aim of this book is to present these results in a coherent way. We start from the beginning, namely Bernstein's beautiful theorem which expresses the number of solutions of generic systems in terms of the mixed volume of their Newton polytopes. We give complete proofs, over arbitrary algebraically closed fields, of Bernstein's theorem and its recent extension to the affine space, and describe some open problems. We also apply the developed techniques to derive and generalize Kushnirenko's results on Milnor numbers of hypersurface singularities which in 1970s served as a precursor to the development of toric geometry. Care was taken to make this book as elementary as possible. In particular, we develop all the necessary algebraic geometry (modulo some explicitly stated basic results).

研究动机与目标

• 为从环面到仿射空间 kⁿ 的 Bernstein 定理扩展提供完整且连贯的处理。
• 根据其 Newton 多面体和混合体积，表征多项式方程的一般系统。
• 提供 Bernstein 定理及其近期扩展的初等、自包含证明，使其在任意代数闭域上均可访问。
• 利用所发展的框架，推广 Kushnirenko 关于超曲面奇点 Milnor 数的结果。
• 识别并讨论使用几何方法枚举多项式组解时的开放问题。

提出的方法

• 利用 торическая 几何分析多项式组在无穷远处的行为，特别关注 Newton 多面体的几何结构。
• 应用混合体积理论，计算仿射空间中一般系统孤立解的个数。
• 系统研究环面的紧化及其边界除子，以理解环面之外的解数。
• 从零开始建立基础代数几何概念，确保可及性的同时保持数学严谨性。
• 利用 Newton 多面体的结构，定义并表征多项式组的通用性条件。
• 建立 Newton 多面体的混合体积与解数（按重数计数）之间的精确对应关系。

实验结果

研究问题

• RQ1 如何将 Bernstein 定理中通过混合体积计数解的方法，从环面扩展到整个仿射空间 kⁿ？
• RQ2 Newton 多面体的何种几何条件可确保多项式组在 kⁿ 中仅有有限多个孤立解？
• RQ3 解在无穷远处的行为如何影响仿射空间中解的总数？
• RQ4 Newton 多面体与混合体积以何种方式表征代数闭域上多项式组的通用性？
• RQ5 如何利用 торическая 几何与解计数的框架，推广 Kushnirenko 关于 Milnor 数的结果？

主要发现

• 在代数闭域上，n 个多项式在 n 个变量中的孤立解的个数，由其 Newton 多面体的混合体积给出。
• Bernstein 定理向仿射空间的扩展已完全解决，为 kⁿ 中的解数提供了完整公式。
• 通用性通过 Newton 多面体的相对位置与结构进行几何表征，确保在无穷远处的横截性。
• 该方法对解计数提供了统一处理，同时涵盖环面解与仿射解，统一了先前的结果。
• 该框架通过将 Milnor 数计算与混合体积及解计数相联系，推广了 Kushnirenko 的工作。
• 所有结果均在任意代数闭域上证明，书中提供了完整的理论发展。