Cubic-degree lower bound for the oscillation index

Prove that the generalized lower bound \(\sigma_{f,Z}\geq (n-s(f))/3\) holds for every non-constant polynomial \(f\in K[x_1,\ldots,x_n]\) of degree 3, where \(s(f)\) denotes the dimension of the singular locus determined by the highest-degree homogeneous part and \(Z\) is the specified subscheme.

Background

The paper introduces generalized log-canonical-threshold bounds for the oscillation index $\sigma_L(f,Z)$, which controls exponential sums over non-Archimedean local fields. A conjectural estimate in the cited literature predicts a lower bound involving $(n-s(f))/\deg(f)$.

The authors show that their invariant $\sigma_{f,Z}$ cannot satisfy this bound in complete generality, but they verify equality in an important case: homogeneous cubic polynomials with $s(f)=0$. They then formulate the cubic-degree inequality as a concrete unresolved question, motivated in part by possible improvements to results on the cubic local-global principle.