# Symbolic powers of monomial ideals

## TL;DR

This paper studies the asymptotic behavior of the lengths of certain symbolic powers of monomial ideals, showing that specific coefficients in their quasi-polynomial expressions become constant under particular conditions.

## Contribution

It proves that the second leading coefficient in the quasi-polynomial describing lengths of symbolic powers stabilizes to a constant when the ideal is generated by elements of the same degree and has height at least 2.

## Key findings

- The function describing lengths is of quasi-polynomial type with periodic coefficients.
- The leading coefficient is constant for large n, as previously shown.
- The second coefficient also becomes constant under specified conditions.

## Abstract

Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n \gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i = 0, \ldots, c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c \neq 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for $n \gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I \geq 2$ then $a_{c-1}(-)$ is also a constant.