Title: Nonlinear Loewy Factorizable Algebraic ODEs and Hayman's Conjecture,
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.