Nonlinear Loewy Factorizable Algebraic ODEs and Hayman's Conjecture

TL;DR

This paper introduces a new class of nonlinear algebraic ODEs, analyzes their meromorphic solutions' growth, and confirms Hayman's conjecture for second-order cases, showing most solutions are elliptic or degenerate elliptic functions.

Contribution

It defines nonlinear Loewy factorizable algebraic ODEs, studies their solutions' growth, and verifies Hayman's conjecture for second-order equations.

Findings

Meromorphic solutions are elliptic or degenerate elliptic functions.

Most solutions have order of growth at most two.

Hayman's conjecture holds for these second-order ODEs.

Abstract

In this paper, we introduce certain $n$-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth are at most two. Moreover, for the second order factorizable algebraic ODEs, all the meromorphic solutions of them (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.