Lorentzianity of tropical volume polynomials

Determine for which tropical varieties in a product of real vector spaces and which sequences of positive tropical divisors the associated tropical volume polynomial is Lorentzian.

Background

The paper proves that projection-purity and facet-selectability determine the support of tropical multidegrees through a polymatroid base polytope, but it also constructs examples showing that these conditions do not guarantee Lorentzianity of the tropical volume polynomial. The failure persists even for translation-admissible tropical varieties connected in codimension one.

The authors therefore ask for a characterization of the tropical varieties and positive tropical divisors for which Lorentzianity does hold. Augmented Bergman fans of polymatroids are established as one class satisfying this property, but the broader classification remains unresolved.

References

This contrast leads to a general question:

For which tropical varieties $ \Gamma\subset N_R=R{m_1}\times\cdots\timesR{m_p} $ and which positive tropical divisors $\underline\Lambda=\Lambda_1,\ldots,\Lambda_p$, $\Lambda_i\subsetR{m_i}$, is the tropical volume polynomial $ {\rm tvol}_{\Gamma,\underline\Lambda}() $ Lorentzian?

— When are tropical multidegrees positive?  (2608.25987 - Cid-Ruiz, 26 Aug 2026) in Section 1, immediately following the discussion of the non-Lorentzian counterexamples