AI 中文总结
该研究针对多级通用初始理想的正则性问题，基于特定单项式序证明了R/I正则性的下界，并将其应用得出d-Leray单纯复形的最优彩色分数Helly定理。
AI 中文摘要
Bayer和Stillman在1987年的著名结果指出，对于多项式环S的齐次理想I，在逆字典式单项式序下，S/I与S/GIN(I)的正则性相同，其中GIN(I)是通用初始理想。若多项式环R的变量被划分为互不相交的变量块X₁,…,X_c，则R上存在自然的多级分次，可类似地为R的任何多重齐次理想I定义多级通用初始理想。然而，Bayer-Stillman定理的全部效力在多级分次情形下失效：存在多重齐次理想I，无论选择何种单项式序，在过渡到多级通用初始理想后正则性均不被保留。我们基于I的多级通用初始理想在每个变量块上的限制的几乎正则序列，证明了R/I正则性的下界。再次使用逆字典式单项式序，但有趣的是，该下界结果要求对变量采用特定的序作为条件。作为应用，我们证明了d-Leray单纯复形的最优分数Helly定理，该问题源自Kim在2017年的工作。
英文摘要
A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer--Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.
