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关。这项研究的中心主题是符号幂，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。这个项目旨在解决两个问题，一是比较符号幂的几何概念与自然幂的代数概念，二是p-导子的微分代数概念在交换代数中的应用。研究人员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研究理想的象征力量。研究主要集中在两个问题上：包含问题和p-导子在交换代数中的应用。当多项式环中的理想对应于在仿射空间中的某一簇上消失的多项式时，它们的符号幂由在给定簇上消失到某一阶的多项式组成。这个自然几何概念的代数描述来自于初等分解理论，这是算术基本定理的理想理论版本。这是一个古老的、丰富的、无处不在的话题，然而，关于象征性权力的问题很多，既容易表达，又很难解决。遏制问题试图将符号幂，一种自然的几何概念，与普通的幂，一种自然代数概念进行比较。这涉及到其他有趣的问题，例如确定多项式在给定簇上消失的最小次数。该项目的另一个部分是发现p-导子和交换代数之间的新联系，着眼于一种新的混合特征奇点理论，它本着F-奇点理论及其与D-模的联系的精神，将p-导子与微分算子相结合。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.