Asymptotic behaviour of integer programming and the $\text{v}$-function of a graded filtration

TL;DR

This paper establishes that the $\\operatorname{v}$-function associated to a Noetherian graded filtration is eventually quasi-linear, extending known results for ordinary and integral closures of powers to a broad class of filtrations. The authors develop a framework using stabilization of associated primes, Ratliff-type colon properties, and Koszul homology to prove linear behavior of $\\operatorname{v}_{\\mathfrak p}(I_{[k]})$ for primes in $\\operatorname{Ass}^{\\infty}(\\mathcal{I})$, which together imply quasi-linearity of $\\operatorname{v}(I_{[k]})$. They further connect the computation of $\\operatorname{v}$-numbers for monomial filtrations to asymptotic integer programming, providing equivalences among solvability of programs, associated primes, and homological invariants, and demonstrate these ideas concretely via the Macaulay2 package \\texttt{VNumberFSPack}. The work culminates with open questions on the relation between $\\operatorname{v}$-numbers and Castelnuovo-Mumford regularity, stability indices, and non-Noetherian cases, outlining future directions in both theory and computational tools.

Abstract

The $\text{v}$-function of a graded filtration $\mathcal{I}=\{I_{[k]}\}_{k\ge0}$ is introduced. Under the assumption that $\mathcal{I}$ is Noetherian, we prove that the $\text{v}$-function $\text{v}(I_{[k]})$ is an eventually quasi-linear function. This result applies to several situations, including ordinary powers, and integral closures of ordinary powers, among others. As another application, we investigate the asymptotic behaviour of certain integer programming problems. Finally, we present the \textit{Macaulay2} package $\texttt{VNumber}$.

Theorems & Definitions (19)

• Lemma 1.1
• proof
• Remark 1.2
• Theorem 1.3
• Lemma 1.4
• Lemma 1.5
• Proposition 1.6
• proof
• proof : Proof of Lemma \ref{['Lem:FS24-2']}
• proof : Proof of Theorem \ref{['Thm:FS24']}