Symbolic Powers and p-Derivations

这是一个交换代数的项目，与代数几何、组合学和算术几何有联系。研究的中心是符号幂的主题，这是一个活跃的研究课题，与交换代数的几乎所有方面都有联系。本项目旨在解决两个问题，一个是符号幂的几何概念与自然幂的代数概念的比较，另一个是p导的微分代数概念在交换代数中的应用。研究者将指导本科生的研究经验，并组织一个交换代数的研究生研讨会。这个项目的统一主题是对理想的象征力量的研究。研究主要集中在两个问题：包含问题和p导在交换代数中的应用。多项式环上的理想对应于仿射空间中在某一变量上消失的多项式，而它们的符号幂则由在给定变量上以某一阶消失的多项式组成。这个自然的几何概念有一个来自初等分解理论的代数描述，这是算术基本定理的一个理想理论版本。这是一个古老的、丰富的、无处不在的话题，然而有大量关于象征性权力的问题，这些问题既容易表述，又很难解决。遏制问题试图比较符号幂（一种自然的几何概念）和普通幂（一种自然的代数概念）。这与其他有趣的问题有关，例如确定多项式在给定变量上消失的最小度。项目的另一部分是发现p导与交换代数之间的新联系，着眼于在混合特征中结合p导与微分算子的新奇点理论，其精神是f -奇点理论及其与d -模的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. The research centers on the theme of symbolic powers, an active topic of research with connections to virtually all aspects of commutative algebra. This project aims to resolve two questions, on comparing the geometric notion of symbolic powers with the algebraic notion of natural powers, and on the application of the differential algebraic notion of p-derivations to commutative algebra. The investigator will direct undergraduate research experiences and organize a graduate workshop in commutative algebra.The unifying theme of this project is the study of symbolic powers of ideals. The research focuses primarily on two questions: the containment problem and applications of p-derivations to commutative algebra. While ideals in a polynomial ring correspond to the polynomials that vanish on a certain variety in affine space, their symbolic powers consist of the polynomials that vanish to a certain order on the given variety. This natural geometric notion has an algebraic description coming from the theory of primary decomposition, an ideal-theoretic version of the fundamental theorem of arithmetic. This is an old, rich, and ubiquitous topic, yet there is an abundance of questions about symbolic powers that are both easy to phrase and very difficult to solve. The containment problem attempts to compare symbolic powers, a natural geometric notion, with ordinary powers, a natural algebraic notion. This relates to other interesting questions, such as determining the minimal degrees of polynomials vanishing on a given variety. Another part of the project is to discover new connections between p-derivations and commutative algebra, with an eye towards a new singularity theory in mixed characteristic that combines p-derivations with differential operators in the spirit of the theory of F-singularities and its connections to D-modules.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.