Axiomatic generalization of characteristic-polynomial properties

Determine whether the properties of the characteristic polynomial of a (q,r)-polymatroid that involve independent spaces, circuits, dependent spaces, or independent spaces can be generalized to the corresponding structures of rational q-polymatroids.

Background

The paper introduces the characteristic Puiseux polynomial for rational q-polymatroids and observes that lattice-theoretic results for characteristic polynomials of (q,r)-polymatroids can often be transferred to this setting. However, properties involving combinatorial structures such as circuits, dependent spaces, and independent spaces require notions specific to rational q-polymatroids.

The authors explicitly state that it is currently unknown whether those properties admit analogous formulations in the rational q-polymatroid setting. Resolving this would establish a broader theory for the characteristic Puiseux polynomial.