A tropical framework for using Porteous formula

Given a rational polyhedral space $X$ (a tropical cycle with boundary, in the sense of Mikhalkin–Rau), one can define tropical vector bundles on $X$ having real or tropical fibers. By restricting attention to bounded rational sections of these bundles, one obtains characteristic classes that behave as expected classically. We develop further properties of these classes and use them to prove a tropical analogue of the splitting principle, which allows us to establish the foundations for Porteous’ formula in this setting: a determinantal expression for the fundamental class of degeneracy loci in terms of Chern classes. The boundary framework is essential, as it allows the rank of a bundle morphism to drop at sedentary strata, giving degeneracy loci their expected codimension.

💡 Research Summary

The paper develops a comprehensive tropical analogue of Porteous’ formula by introducing tropical vector bundles on rational polyhedral spaces with boundary, as defined by Mikhalkin–Rau. The author first extends the usual tropical cycle framework from the interior of ℝⁿ to the extended tropical affine space Tⁿ = (ℝ ∪ {−∞})ⁿ, allowing faces to meet the “boundary” where coordinates become −∞. Points are equipped with a “sedentarity” set I(p) recording which coordinates are −∞; the corresponding stratum Tⁿ_I is identified with ℝ^{n−|I|}. This boundary structure is crucial because it permits the rank of a bundle morphism to drop when entries tend to −∞, thereby creating degeneracy loci of the expected codimension.

Tropical vector bundles are defined with either real or tropical fibers. The author restricts to bounded rational sections—sections that remain finite on each sedentarity stratum—so that characteristic classes can be defined exactly as in classical algebraic geometry. Chern classes cₖ(E) are introduced via a Chern polynomial c_t(E)=∑cₖ(E)tᵏ, and they behave functorially under pull‑back, tensor product, and direct sum.

A central technical achievement is the tropical splitting principle. The paper develops a tropical linear algebra in which the invertible matrices are precisely those in the set G(n) (one finite entry per row). The tropical determinant trop det(A) is finite if and only if A∈G(n), and the tropical rank is defined by the usual minor‑vanishing condition (with −∞ playing the role of zero). By conjugating a matrix with permutation and diagonal matrices in G(n), the rank is shown to be invariant and equal to the number of columns containing a finite entry. Using these tools, any tropical vector bundle can be “split” after pulling back to a suitable refinement, becoming a direct sum of tropical line bundles. Consequently, the Chern polynomial of a bundle factors as the product of the Chern polynomials of its line summands, exactly mirroring the classical splitting principle.

With these foundations, the author addresses the main problem: given a morphism ϕ : E → F of tropical bundles over a rational polyhedral space X, with bounded matrix entries, define the degeneracy locus D₀(ϕ) = {x ∈ X | rank ϕ(x)=0}. Because the entries may become −∞ on boundary strata, the rank can drop precisely where the sedentarity increases, and D₀(ϕ) acquires the expected codimension (e·f) where e=rank E and f=rank F. The paper proves the tropical Porteous formula in the rank‑zero case: