Time: 14h00 – 15h, Tuesday October 22, 2024

Venue: Vietnam Institute for Advance in Mathematics (VIASM)

Abstract: A density function for an algebraic invariant is a measurable function on the set of real numbers which measures the invariant on a real scale. In this talk, we will discuss density functions for Noetherian filtrations of homogeneous ideals in a standard graded Noetherian domain over a field $k$. This was inspired by the Hilbert-Kunz density functions developed by V. Trivedi. We will also demonstrate a density function for a finitely generated bigraded module over a bigraded Noetherian $k$-algebra, which is generated in bidegrees $(1,0), (d_1, 1), ldots, (d_r,1)$ for some $d_i geq 0$. Our main ingredients are the method of Gr"obner bases and Sturmfels' structure theorem for vector partition functions. As an application, we will provide a new numerical criterion for integral dependence of arbitrary homogeneous ideals in terms of computable and well-studied invariants, such as, mixed multiplicities of ideals and Hilbert-Samuel multiplicities of certain standard graded algebras. A novelty of our approach is that it does not involve localizations. This talk is based on joint works with Suprajo Das and Vijaylaxmi Trivedi..

Program of Special Semester on Commutative Algebra