Binomial expansion and the $\mathrm{v}$-number

TL;DR

This paper investigates how the $ ext{v}$-function behaves under binomial expansion for monomial ideals, providing explicit formulas for symbolic powers and integral closures, advancing understanding of their algebraic properties.

Contribution

It introduces explicit formulas for the $ ext{v}$-function of sums of monomial ideals' symbolic powers and their integral closures, extending previous knowledge in algebraic combinatorics.

Findings

Derived an explicit formula for $ ext{v}((I+J)^{(k)})$ in terms of $ ext{v}(I^{(i)})$ and $ ext{v}(J^{(j)})$

Extended formulas to the $ ext{v}$-function of integral closures of $(I+J)^k$

Analyzed the behavior of the $ ext{v}$-function under binomial expansion for monomial ideals

Abstract

Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under binomial expansion. As an application, we get an explicit formula of $\mathrm{v}((I+J)^{(k)})$ in terms of $\mathrm{v}(I^{(i)})$ and $\mathrm{v}(J^{(j)})$, where $L^{(k)}$ denote the symbolic power of an ideal $L$. Furthermore, an analogous formula is extended for the $\mathrm{v}$-function of integral closure of $(I+J)^k$.