On the density of Second Order Differential Equations without algebraic solutions on $\mathbb{P}}^{2}$

TL;DR

This paper proves that most second order differential equations of high enough degree in the projective plane lack algebraic solutions, extending the result to webs on the same space.

Contribution

It establishes the generic non-existence of algebraic solutions for high-degree second order differential equations and webs on the projective plane.

Findings

Generic second order differential equations of large degree have no algebraic solutions.

The result extends to webs on the projective plane.

Provides conditions under which algebraic solutions do not exist.

Abstract

We prove that a generic second order differential equation in the projective plane has no algebraic solutions when the bidegree is big enough. We also proof an analogous result for webs on $\mathbb{P}}^{2}$.