从最小自由分解的角度看Eisenbud-Green-Harris猜想

Eisenbud-Green-Harris猜想是交换代数中的一个重要猜想。它刻画了多项式环中一类分次理想的希尔伯特函数。在猜想中，我们假定这些分次理想包含了由给定次数的齐次多项式所构成的正则列。研究希尔伯特函数常用的工具是Grobner基理论，然而这一工具在处理正则列时并不合适。在本项目中，我们计划从最小自由分解的角度来研究这个关于希尔伯特函数的猜想。从这个新的角度出发，一方面，我们可以把这个猜想和Lex-Plus-Powers猜想联系起来；另一方面，我们可以探索Boij-Soderberg理论在这个猜想上会发挥多大的作用。更具体地说，我们会重新研究一些关于此猜想的已知的结果和例子，并试着从中找出一般性的想法，然后希望能得到一些关于此猜想的新结果。Eisenbud-Green-Harris 猜想的解决将不仅仅是交换代数中的一大突破，还会影响到代数几何和组合数学等领域。

The Eisenbud-Green-Harris Conjecture is an important conjecture in Commutative Algebra. It characterizes the Hilbert functions of a class of graded ideals in the polynomial ring. These graded ideals are supposed to contain regular sequences of homogeneous polynomials of specific degrees. The usual tool for studying Hilbert functions is the Grobner basis theory. However, this tool is not efficient for dealing with regular sequences. In this project, we plan to study this conjecture from the minimal free resolution point of view. With this new point of view, on one hand, we can relate this conjecture to the Lex-Plus-Powers Conjecture; on the other hand, we can explore the use of the Boij-Soderberg theory on this conjecture. More specifically, we will go over some known results and examples about this conjecture and try to extract some general ideas from them, and then hopefully, we can get some new results about this conjecture. If the Eisenbud-Green-Harris Conjecture is solved, then it will not only be a breakthrough in Commutative Algebra, but also have an impact on other fields like Algebraic Geometry and Combinatorics.