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

Abstract

The $\text{v}$-function of a graded filtration $\mathcal{I}=\{I_{[k]}{\}}_{k\ge 0}$ 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 $\text{VNumber}$.

This is the final version of our paper, accepted for publication in Journal of Algebra and its Applications