A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group
Alexander Lazar

TL;DR
This paper proves a conjecture relating the tropical Picard groups of two graphs and their product, using concepts from tropical geometry, chip-firing games, and algebraic structures on tropical complexes.
Contribution
It establishes the injectivity and surjectivity of a map between the Picard groups of graphs and their product, confirming Cartwright's conjecture.
Findings
Proved the conjecture relating Picard groups of graphs and their product.
Analyzed properties of tropical complexes and their relation to chip-firing games.
Established conditions under which the map between Picard groups is surjective.
Abstract
In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex is a -complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors that are locally principal are called Cartier. Two divisors that differ by a principal divisor are said to be linearly equivalent. The linear equivalence classes of Cartier divisors on a tropical complex form a group called the Picard group of , by analogy to the definition of the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
