Newton polygons and curve gonalities

TL;DR

This paper establishes a combinatorial upper bound for the gonality of algebraic curves defined by Laurent polynomials, linking geometric properties to Newton polygons and graph theory, and conjectures its generic attainability.

Contribution

It introduces a new combinatorial upper bound for curve gonality based on Newton polygons and proves it in several special cases, connecting algebraic geometry with graph theory.

Findings

Proposed a combinatorial upper bound for gonality

Reduced the conjecture to a combinatorial statement using graph theory

Proved the bound in multiple special cases

Abstract

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.