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

Abstract: The $\text{v}$-function of a graded filtration $\mathcal{I}={I_{[k]}}{k\ge0}$ 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 $\texttt{VNumber}$.