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DTSTART;TZID=Asia/Seoul:20261006T163000
DTEND;TZID=Asia/Seoul:20261006T173000
DTSTAMP:20260821T005558Z
CREATED:20260821T005539Z
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UID:13133-1791304200-1791307800@dimag.ibs.re.kr
SUMMARY:Daniel McGinnis\, Multi-generic initial ideals\, regularity\, and the optimal colorful fractional Helly theorem for $d$-Leray complexes
DESCRIPTION: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. \nWe 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. \nAs an application\, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes\, a problem stemming from the work of Kim in 2017.
URL:https://dimag.ibs.re.kr/event/2026-10-06/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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