Translation-admissibility of augmented Bergman fans

Determine whether the augmented Bergman fan associated with every polymatroid is a translation-admissible tropical variety, equivalently whether its Minkowski sum with every rational linear subspace is pure-dimensional.

Background

The paper proves that augmented Bergman fans of polymatroids are projection-pure and facet-selectable, and consequently have positive tropical multidegrees supported on the lattice points of the corresponding polymatroid base polytope. It also proves that their tropical volume polynomials are Lorentzian.

Translation-admissibility is a stronger property than the two conditions used in the paper. Although every translation-admissible tropical variety satisfies projection-purity and facet-selectability, the converse is false in general. The authors explicitly note that their results do not determine whether augmented Bergman fans themselves satisfy translation-admissibility.

References

Although \autoref{thmC} places every augmented Bergman fan within the scope of \autoref{thmB}, it does not settle whether augmented Bergman fans are translation-admissible. We are therefore led to the following natural question:

Let $\Sigma_P\subset\prod_{i=1}pR{E_i}$ be the augmented Bergman fan associated with a polymatroid $P$. Is the augmented Bergman fan $\Sigma_P$ a translation-admissible tropical variety? Equivalently, is $\Sigma_P+V$ pure-dimensional for every rational linear subspace $V\subseteq\prod_{i=1}pR{E_i}$?

— When are tropical multidegrees positive?  (2608.25987 - Cid-Ruiz, 26 Aug 2026) in Section 1, immediately after the corollary on polymatroid base-polytope realizability