Graded Syzygies: Geometry and Computation

分级 Syzygies：几何和计算

• 批准号: 1900792
• 负责人: Jason McCullough
• 金额: $ 15万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900792&HistoricalAwards=false
• 关键词: Graded; Syzygies; Geometry; Computation

This research focuses on several problems in computational commutative algebra and algebraic geometry. This project is concerned with the study of systems of polynomial equations in several variables. Polynomial equations model many real world phenomena, ranging from robotics motion planning to physics to conservation programs. To each system of polynomial equations, we associate a set of points, called a variety, which correspond to the set of solutions for all the equations. There is a growing body of literature that shows that the computational complexity of the system of polynomial equations has deep connections to the geometry of the associated variety. In nice cases, when the associated variety is smooth, there are well understood bounds limiting the computational complexity. In the worst possible cases, complexity is doubly exponential. Recent work of the PI shows there is a middle ground which is still poorly understood. The goal of this research is to better understand this connection.The research aims to attack several open questions concerning projective bounds on syzygies. One goal is to study Rees-Like Algebras, which were essential in the PI's construction of counterexamples to the Eisenbud-Goto Conjecture (joint with I. Peeva), and to relate their properties to the more well-studied Rees Algebras. A second goal is to provide effective bounds on invariants, such as the Castelnuovo-Mumford regularity or the degrees of individual syzygies, in terms of other invariants. All of the projects have a computational flavor and include writing code for Macaulay2, an NSF-sponsored computer algebra system maintained by Grayson and Stillman. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

本文主要研究计算交换代数和代数几何中的几个问题。本课题主要研究多变量多项式方程组。多项式方程模拟了许多真实世界的现象，从机器人运动规划到物理学再到守恒程序。对于每一组多项式方程，我们将一组点关联起来，称为变化，它们对应于所有方程的解的集合。有越来越多的文献表明，多项式方程组的计算复杂性与相关变种的几何有着深刻的联系。在很好的情况下，当相关的变化是光滑的时，存在众所周知的限制计算复杂性的界限。在可能的最坏情况下，复杂性是双指数的。国际和平研究所最近的工作表明，存在一个仍然鲜为人知的中间立场。这项研究的目的是为了更好地理解这种联系。这项研究的目的是解决关于合子的投影界的几个公开问题。一个目的是研究Rees-Like代数，它是PI构造Eisenbud-Goto猜想反例(与I.Peeva联合)所必需的，并将它们的性质与更好地研究的Rees代数联系起来。第二个目标是根据其他不变量提供不变量的有效界，例如Castelnuovo-Mumford正则性或单个合子的次数。所有的项目都有计算的味道，包括为Macaulay2编写代码，Macaulay2是由NSF赞助的计算机代数系统，由Grayson和Stillman维护。该项目由代数和数论计划和既定的激励竞争性研究计划(EPSCoR)共同资助。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。

项目成果

The Regularity Conjecture for prime ideals in polynomial rings

• DOI: 10.4171/emss/38
• 发表时间: 2021-01
• 期刊: EMS Surveys in Mathematical Sciences
• 影响因子: 2.3
• 作者: J. McCullough;I. Peeva

On the maximal graded shifts of ideals and modules

论理想与模的最大分级位移

• DOI: 10.1016/j.jalgebra.2018.09.037
• 发表时间: 2021
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: McCullough, Jason

Canonical Modules and Class Groups of Rees-Like Algebras

类Rees代数的规范模和类群

• DOI: 10.1307/mmj/20205974
• 发表时间: 2023
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Mantero, Paolo;McCullough, Jason;Miller, Lance Edward
• 通讯作者: Miller, Lance Edward

G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras

外代数的 G 二次商、LG 二次商和 Koszul 商

• DOI: 10.1080/00927872.2022.2029875
• 发表时间: 2022
• 期刊: Communications in Algebra
• 影响因子: 0.7
• 作者: McCullough, Jason;Mere, Zachary
• 通讯作者: Mere, Zachary

Singularities of Rees-like algebras

类里斯代数的奇点

• DOI: 10.1007/s00209-020-02524-6
• 发表时间: 2020
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Mantero, Paolo;Miller, Lance Edward;McCullough, Jason
• 通讯作者: McCullough, Jason