Characteristic Puiseux polynomials beyond the special paving case

Describe the characteristic Puiseux polynomial for rational convex combinations of q-matroids beyond the special case of equally ranked paving q-matroids, characterize its general properties, and determine whether it can be expressed solely in terms of 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 arising from a particular construction and having the same rank. The authors emphasize that this covers only a restricted situation.

The unresolved questions concern three extensions: obtaining descriptions in other cases, understanding the general properties of the invariant, and determining whether the invariant is recoverable solely from the characteristic polynomials of the constituent q-matroids.

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