Poincar\'e Series of Divisors on Graphs and Chains of Loops

TL;DR

This paper proves the rationality of Poincaré series for divisors on finite graphs and chains of loops, providing algorithms for their computation based on algebraic and combinatorial methods.

Contribution

It introduces new proofs of rationality for Poincaré series on graphs and chains of loops, and develops algorithms for their explicit calculation.

Findings

Proves rationality of Poincaré series for both graphs and chains of loops.

Develops algorithms for computing Poincaré series in these settings.

Uses homomorphisms and lattice point enumeration techniques.

Abstract

We study Poincar\'e series associated to a finite collection of divisors on i. a finite graph and ii. a certain family of metric graphs called chain of loops. Our main results are proofs of rationality of the Poincar\'e series and algorithms for computing it in both these cases. The main tools used in the proof of rationality are the following. For graphs, we study a certain homomorphism from a free Abelian group of finite rank to the direct sum of the Jacobian of the graph and the integers. For chains of loops, our main tool is an analogue of Lang's conjecture for Brill-Noether loci on a chain of loops and adapts the proof of rationality of the Poincar\'e series of divisors on an algebraic curve (over an algebraically closed field of characteristic zero). Our algorithms are based on a closer study of the objects involved in the proof of rationality, for instance, computing the fibres of certain homomorphisms and lattice point enumeration in rational polyhedra.