An exact test built from projections
A new mathematical preprint reports an exact test for when a tropical multidegree is positive, but only for tropical varieties that meet two structural conditions: projection-purity and facet-selectability. Under those conditions, a multidegree is positive exactly when its type satisfies every inequality determined by the dimensions of the variety's natural projections. The result turns positivity into a geometric check: examine the projections, then ask whether all of the required inequalities hold.
The analysis is proof-based and works with rational polyhedral complexes of pure dimension, carrying positive integer weights and satisfying a balancing condition. Its proof strategy combines bounded rational equivalence and recession fans with toric methods based on Minkowski weights, and uses a positivity result as the decisive input. One key reduction takes positivity for the whole tropical variety back to positivity for a facet-parallel linear space, connecting a facet-wise calculation to the global statement.
The projection dimensions do more than provide a checklist. The paper shows that the function assigning a dimension to each projection is the rank function of a polymatroid, a framework that packages the relevant rank constraints. Under the same assumptions, the positive multidegrees are exactly the lattice points in the associated polymatroid base polytope. In that sense, the theorem gives both a test for positivity and a precise description of where positive values can occur.
That result is deliberately conditional. The projection-based criterion applies when projection-purity and facet-selectability hold, rather than to arbitrary tropical varieties. The paper also treats this positivity criterion as separate from the question of whether the associated tropical volume polynomial is Lorentzian.
Positive values do not settle the polynomial question
The paper uses "Lorentzian" as a separate property of the tropical volume polynomial and examines it through Hessian eigenvalue calculations. A constructed example shows why the distinction matters. Its displayed tropical volume coefficients are 10, 6, 6 and 4, yet the paper concludes that the polynomial is non-Lorentzian. Positive coefficients, in other words, do not by themselves settle the separate Lorentzian question in these examples.
A second example is a tropical variety connected in codimension one. Its reported multidegrees are 5, 4 and 5, while the displayed tropical volume polynomial has coefficients 5, 8 and 5. The Hessian calculation reports eigenvalues of 2 and 18, and the paper concludes that this polynomial is not Lorentzian.
Together, the examples show that positive multidegree data and a non-Lorentzian volume polynomial can coexist. The two questions must therefore be checked separately in this setting.
Where the two properties meet
A separate theorem identifies a family where the two stories line up: augmented Bergman fans of polymatroids. For an augmented Bergman fan, the paper shows that projection-purity and facet-selectability hold, and that positive multidegrees occur exactly at lattice points of the polymatroid base polytope. The result supplies an exact projection-based description of the support for this family.
For these augmented Bergman fans, the tropical volume polynomial is Lorentzian as well. Taken alongside the counterexamples, that result draws a clear boundary: projection-purity and facet-selectability support the exact positivity criterion, while the Lorentzian conclusion belongs to the augmented Bergman fan setting established by the paper.
With standard tropical hyperplanes, the numerical rule becomes especially simple. A multidegree equals 1 when its index lies inside the polymatroid base polytope and 0 otherwise. The statement gives a direct 0-or-1 description of the multidegree support while retaining the Lorentzian conclusion for the volume polynomial.
A conditional map of an abstract landscape
The work is a proof-based study of mathematical constructions, not an empirical dataset, so it reports no statistical uncertainty. Its conclusions are theorems under stated conditions, and the counterexamples concern selected constructions rather than every tropical variety or every possible divisor sequence. The document is identified as arXiv version 1, dated 26 August 2026. The author reports support from NSF grant DMS-2502321 and Simons Foundation Travel Support for Mathematicians Award MPS-TSM-00013551.
Paper data and sources
Original title: When are tropical multidegrees positive?
Authors: Yairon Cid-Ruiz
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text