Initial behavior of v_p(I^k) and v(I^k): number of strict local maxima

Investigate whether, for a homogeneous ideal I ⊂ S = K[x1,...,xn] and the I-adic filtration, the sequences k ↦ v_p(I^k) and k ↦ v(I^k) can exhibit an arbitrary number of strict local maxima before their eventual linear behavior, where p ranges over the stable associated primes Ass∞(I).

Background

For the I-adic filtration of a homogeneous ideal in a standard graded polynomial ring, v_p(I^k) and v(I^k) are known to be eventually linear. However, the transient behavior before stabilization is largely mysterious.

The authors explicitly state uncertainty about how wild the initial behavior can be, motivating further questions about monotonicity and fluctuation patterns.