On a theorem of Gundersen and Laine. (English) Zbl 0773.34005
Using Nevanlinna’s theory the author studies the algebraic differential equation
\[
\Omega(z,w)=\sum^ n_{k=0}A_ k(z)w^ k \tag{*}
\]
with meromorphic coefficients \(A_ k\) and with a differential polynomial \(\Omega\) of weight \(\Delta\). One of the results is: If \(n>\Delta\) and if \(A_ q\), \(\Delta\leq q\leq n-1\), is a dominant coefficient of exponential type, then equation \((*)\) does not possess a \(\nu\)-valued algebroid solution \(w\) with a small number of branch points. This is a generalization of a theorem of Gundersen and Laine on meromorphic solutions.
Reviewer: F.Gackstatter (Berlin)
MSC:
34M99 | Ordinary differential equations in the complex domain |
30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |