CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM

By Yang An, Matthew Baker, Greg Kuperberg, Farbod Shokrieh. Forum of Mathematics, Sigma, 2013

91 citations recorded by the source corpus.

Abstract Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus $g$ . Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma $ . We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a ‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$ -dimensional real torus $\mathrm{Pic}^g(\Gamma )$ into parallelotopes $C_T$ , one for each spanning tree $T$ of $G$ , and the dual Kirchhoff theorem becomes the statement that the volume of $\mathrm{Pic}^g(\Gamma )$ is the sum of the volumes of the cells in the decomposition.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/27d51817b4861a03a9fdf70c47732b0ad24a148e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. David Jensen
2. Matthew Baker
3. Laura Demarco
4. Omid Amini
5. Lucia Caporaso
6. Robert Rumely

In Computational Theory and Mathematics

1. David Jensen
2. Matthew Baker
3. Omid Amini
4. Lucia Caporaso
5. Dhruv Ranganathan
6. Sam Payne
7. Joseph Rabinoff
8. Melody Chan

In Geometry and Topology

1. David Jensen
2. Matthew Baker
3. Omid Amini
4. Laura Demarco
5. Lucia Caporaso
6. Sam Payne
7. Dhruv Ranganathan
8. Melody Chan

Citations in the corpus

• Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “ [MZ08] utilizes the theory of tropical theta functions and the tropical analogue of Riemann’s theta constant. A purely combinatorial proof based on the theory of q-connected orientations is given in [ABKS14], and the combinatorial proof yields an interesting analogue of Theorem 5.13 for nite graphs”