Asymptotic behavior of integer programming and the v-function of a graded filtration

The [Formula: see text]-function of a graded filtration [Formula: see text] is introduced. Under the assumption that [Formula: see text] is Noetherian, we prove that the [Formula: see text]-function [Formula: see text] 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 behavior of certain integer programming problems. Finally, we present the Macaulay2 package [Formula: see text].