On implicit ODEs with hexagonal web of solutions

TL;DR

This paper classifies implicit ODEs with hexagonal webs of solutions, focusing on the case where the web's curvature vanishes, and establishes normal forms under natural regularity assumptions.

Contribution

It provides a finite list of normal forms for implicit ODEs with hexagonal solution webs, assuming smoothness conditions, advancing the geometric understanding of such equations.

Findings

Normal forms for hexagonal 3-webs of solutions are established.

The curvature form vanishes identically in the symmetric case.

Regularity assumptions ensure smoothness of the defining surface and critical set.

Abstract

Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometrical meaning of these assumptions is that the surface, defined by ODE in the space of 1-jets, is smooth as well as the criminant, which is the critical set of this surface projection to the plane.