Applying the Frobenius Morphism and Convexity to Study Singularities

本课题主要研究有理数域上某些多项式系统的解的集合。多项式是纯数学和应用数学中最基本和最重要的函数之一，它是通过对一组变量迭代加法和乘法的基本函数而产生的。多项式方程的解集可以用几何来解释，自然界中出现的许多形状，如球体和椭圆形，都可以用它们来描述。在纯数学和应用数学中，以及在统计学、工程学和计算机科学中，多项式和它们所描述的几何对象是无处不在的。这个方案的一个重要主题是研究定义在有理数上的多项式方程，通过归一化为正特征。我们的方法很大程度上涉及到Frobenius态射的使用，但也依赖于凸几何的基本概念(例如，圆锥和多面体)。更确切地说，我们建议1)利用Frobenius和凸性来研究Bernstein-Sato多项式；2)利用凸性和积分来研究正特征的某些重性(例如F-签名和Hilbert-kunz重数)；以及3)研究理想的广义Frobenius幂，并将它们与检验理想和乘子理想联系起来。

This project concerns the study of the set of solutions to certain systems of polynomials over the rational numbers. Polynomials are among the most basic and important functions in pure and applied mathematics, and come from iterating the basic functions of addition and multiplication on a set of variables. The set of solutions to polynomial equations can be interpreted geometrically, and many of the shapes appearing in nature such as spheres and ellipses, can be described using them. Polynomials and the geometric objects they describe are ubiquitous, both in pure and applied mathematics, as well as in statistics, engineering, and computer science. An important theme in this proposal is to study polynomial equations defined over the rational numbers via reduction to positive characteristic. Much of our approach involves the use of the Frobenius morphism, but also relies on basic notions from convex geometry (e.g., cones and polytopes). More precisely, we propose to 1) use Frobenius and convexity to study Bernstein-Sato polynomials; 2) use convexity and integration to investigate certain multiplicities in positive characteristic (e.g. F-signature and Hilbert-Kunz multiplicity); and 3) study the generalized Frobenius powers of an ideal, and relate them to test and multiplier ideals.