Generalized multiplicities of edge ideals

TL;DR

This paper investigates the relationship between generalized multiplicities of square-free monomial ideals and hypergraph structures, providing formulas, bounds, and computations for these invariants using algebraic and geometric methods.

Contribution

It establishes multiplicativity of the $j$-multiplicity over hypergraph components and relates it to Hilbert-Samuel multiplicity, offering new computational tools.

Findings

$j$-multiplicity is multiplicative over hypergraph components

Explicit relation between $j$-multiplicity and Hilbert-Samuel multiplicity

Derived bounds and computed invariants for uniform hypergraphs

Abstract

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the $j$-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the $j$-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.