The $\text{v}$-function of powers of sums of ideals

TL;DR

This paper investigates the behavior of the v-function for powers of sums of ideals in polynomial rings, providing explicit formulas for monomial ideals and analyzing how the v-function interacts with ideal sums.

Contribution

It introduces a detailed analysis of the v-function for sums of ideals, including explicit formulas for monomial ideals, advancing understanding of ideal powers in polynomial rings.

Findings

Explicit formula for v(L^k) when I and J are monomial ideals

Analysis of v-function behavior under ideal sum operations

Insights into the structure of powers of sums of ideals

Abstract

Let $K$ be a field, $I\subset R=K[x_1,\dots,x_n]$ and $J\subset T=K[y_1,\dots,y_m]$ be graded ideals. Set $S=R\otimes_KT$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions $\text{v}(I^k)$ and $\text{v}(J^k)$ is investigated. When $I$ and $J$ are monomial ideals, we describe $\text{v}(L^k)$, giving an explicit formula involving $\text{v}(I^k)$ and $\text{v}(J^k)$.