Related papers: Kirchhoff's theorem for Prym varieties

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are…

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared…

We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show…

For any closed orientable 3-manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to…

The $k$-matching polytope of a graph is the convex hull of all its matchings of a given size $k$ when they are considered as indicator vectors. In this paper, we prove that the $k$-matching polytope of a bipartite graph is normal, that is,…

We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give…

We give a purely tropical analogue of Donagi's $n$-gonal construction and investigate its combinatorial properties. The input of the construction is a harmonic double cover of an $n$-gonal tropical curve. For $n = 2$ and a dilated double…

We prove a $q$-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov--Witten invariants with a $\lambda_g$ class in toric surfaces. Specifically, a generating series of such logarithmic Gromov--Witten…

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The…

Using local detailed balance we rewrite the Kirchhoff formula for stationary distribution of Markov jump processes in terms of a physically interpretable tree-ensemble. We use that arborification of path-space integration to derive a…

We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the…

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit…

Using the method of spectral decimation and a modified version of Kirchhoffs Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely…

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces $\Sigma_n$ and…

We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally…

We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0<r<g-1$) also carries a divisor of degree $2$ and rank $1$. We provide a structure theorem…

Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of $G$. We outline an efficient bijective proof of this theorem, by studying a canonical…

In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally…

The subject of the present paper is phase tropicalization, which was used crucially in the context of Mikhalkin's correspondence theorem for curve counting in the complex coefficient case. The subject can be traced back to Viro's…