Non-Noetherian filtrations: non-quasi-linear v-function and existence of asymptotic slope

Construct a non-Noetherian graded filtration I = {I[k]} of a finitely generated N-graded domain R for which the v-function k ↦ v(I[k]) is not eventually quasi-linear, and determine whether the limit lim_{k→∞} v(I[k])/k exists; if it exists, ascertain its value.

Background

The main theorem establishes eventual quasi-linearity of the v-function under the Noetherian hypothesis on the graded filtration. The authors ask for behavior beyond this hypothesis.

They note that subsequent work [33] provides an example of a non-Noetherian graded filtration with a v-function that is not eventually quasi-linear, but the existence and value of the asymptotic slope lim_{k→∞} v(I[k])/k remain unclear.