Characterize characteristic Puiseux polynomials beyond the special paving case

Characterize characteristic Puiseux polynomials of rational q-polymatroids in situations beyond convex combinations of two paving q-matroids arising from the specified construction, establish their general properties, and determine whether they can be expressed solely through the characteristic polynomials of the summands.

Background

The paper derives an explicit formula for the characteristic Puiseux polynomial of a rational convex combination of two paving q-matroids of equal rank constructed from collections of subspaces satisfying a prescribed intersection condition. This result covers only a special family of rational q-polymatroids. The broader behavior of the invariant, including formulas for other convex combinations and relationships with the characteristic polynomials of the input q-matroids, remains unresolved.

References

This leads to the following questions: Can we describe the characteristic Puiseux polynomial in other situations? Can we say something about its properties, in general? Is it possible to express it only in terms of the characteristic polynomials of the summands?

— The polytope of all $q$-rank functions  (2505.08018 - Alfarano et al., 12 May 2025) in Section 10, item (f) of the Remarks and open questions