Quasiconformal surgery and linear differential equations

TL;DR

This paper introduces a novel quasiconformal surgery method to construct entire functions with specific zero distributions, solving longstanding problems in differential equations and complex analysis.

Contribution

It presents a new construction technique for entire functions solving differential equations with controlled zero distributions, extending previous results and establishing optimal bounds.

Findings

Existence of entire functions of any order > 1/2 with solutions having finite zero exponent of convergence.

Improved bounds on the zeros of solutions to certain differential equations.

Method based on gluing solutions of Schwarzian differential equations.

Abstract

We describe a new method of constructing transcendental entire functions $A$ such that the differential equation $w"+Aw=0$ has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functions $A$ of any prescribed order greater than $1/2$ such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the resulting estimate is best possible. Our method is based on gluing solutions of the Schwarzian differential equation $S(F)=2A$ for infinitely many coefficients $A$.