The lattice size of a lattice polygon

TL;DR

This paper establishes upper bounds on the minimal degrees of models of lattice polygons in projective spaces, linking algebraic geometry with combinatorial properties of Newton polygons, and proves sharpness in generic cases.

Contribution

It provides new bounds on the degrees of models of lattice polygons based on Newton polygon combinatorics, with sharpness results for generic polynomials.

Findings

Upper bounds on minimal degrees in projective models

Sharp bounds for generic Laurent polynomials

Connections between Newton polygons and algebraic models

Abstract

We give upper bounds on the minimal degree of a model in $\mathbb{P}^2$ and the minimal bidegree of a model in $\mathbb{P}^1 \times \mathbb{P}^1$ of the curve defined by a given Laurent polynomial, in terms of the combinatorics of the Newton polygon of the latter. We prove in various cases that this bound is sharp as soon as the polynomial is sufficiently generic with respect to its Newton polygon.