Multiplicity of powers of squarefree monomial ideals

TL;DR

This paper establishes a formula for the multiplicity of powers of squarefree monomial ideals, linking it to associated primes and providing explicit calculations for path ideals of cycles.

Contribution

It introduces a general formula for the multiplicity of powers of squarefree monomial ideals, extending understanding of their algebraic properties.

Findings

Multiplicity of powers is given by a binomial coefficient times the number of associated primes.

The formula applies to all powers, not just initial ones.

Explicit multiplicity calculations for path ideals of cycles are provided.

Abstract

Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. Let $\mu$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of powers of $I$ is given by $$e_0(S/I^s) = \mu \binom{n-d+s-1}{s-1},$$ for all $s \ge 1$. Consequently, we compute the multiplicity of all powers of path ideals of cycles.