WEEKLY ACTIVITIES

Time: 9h30 – 11h00, Wednesday October 23, 2024

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: The notion of epsilon multiplicity, a generalization of the Hilbert-Samuel multiplicity, was introduced by B. Ulrich and J. Validashti to detect integral dependence of arbitrary ideals. This invariant is difficult to handle, as it can be irrational and so the associated length function is very far from being polynomial-like. The objective of this presentation is to discuss certain computational aspects of this invariant. Let $I$ be a homogeneous ideal in a two dimensional standard graded Cohen-Macaulay domain over a field. We will show that the $varepsilon$-multiplicity, $varepsilon(I)$, of $I$ is a rational number. The proof given in arXiv:2402.11935 uses some classical theory in algebraic geometry for projective curves. However, in this talk, we will sketch an alternative proof using purely algebraic methods. We will further describe $varepsilon(I)$ in terms of various mixed multiplicities associated to $I$, which enables us to explicitly compute it using Macaulay2. This talk is based on a joint work with Suprajo Das, Saipriya Dubey, and Jugal K. Verma

Program of Special Semester on Commutative Algebra