# A step to Gronwall's conjecture

## TL;DR

This paper proposes a new approach to prove Gronwall's conjecture by introducing an invariant called the characteristic, which helps identify when a local isomorphism between plane linear 3-webs is a homography.

## Contribution

It introduces the characteristic invariant for generic points of linear 3-webs, advancing the understanding of conditions under which web isomorphisms are homographies.

## Key findings

- The characteristic invariant distinguishes when a web isomorphism is a homography.
- A new method to approach Gronwall's conjecture using invariants.
- Potential progress towards proving Gronwall's conjecture.

## Abstract

In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit an invariant, the {\sl characteristic}, attached to each generic point of such a web, with the following property: if a diffeomorphism interchanges two such linear webs, sending a point of the first to a point of the second which have the same characteristic, then this diffeomomorphism is locally a homography.