Hayman's classical conjecture on some nonlinear second order algebraic ODEs

Nonlinear Loewy Factorizable Algebraic ODEs and Hayman's Conjecture

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 …

On a result of Hayman concerning the maximum modulus set

The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a "small" set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concern…

Picard-Fuchs equations, Integrable Systems, and higher Algebraic K-theory

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.

关于一阶微分方程解的模

On the modulus of solutions of a first order differential equation

本文研究了一阶微分方程解的模估计，证明了解的超阶为n，并给出了二阶代数微分方程解的增长阶描述。