Asymptotic behaviour of the $\text{v}$-number of homogeneous ideals

TL;DR

This paper studies the long-term behavior of the v-number of powers of homogeneous ideals, establishing linear bounds and formulas, and explicitly computes the v-function for certain monomial ideals.

Contribution

It proves that the v-number of large powers of a homogeneous ideal is linear in the power, introducing new blowup algebras and providing explicit calculations for specific cases.

Findings

v-number of powers grows linearly for large exponents

new blowup algebras are constructed for analysis

explicit v-function computed for monomial ideals in two variables

Abstract

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear functions in $k$ are determined for $\text{v}(I^k)$. We call $\text{v}(I^k)$ the $\text{v}$-function of $I$. We prove that $\text{v}(I^k)$ is a linear function in $k$ for $k$ large enough, of the form $\text{v}(I^k)=\alpha(I)k+b$, where $\alpha(I)$ is the initial degree of $I$, and $b\in\mathbb{Z}$ is a suitable integer. For this aim, we construct new blowup algebras associated to graded ideals. Finally, for a monomial ideal in two variables, we compute explicitly its $\text{v}$-function.