Meromorphic solutions of a differential equation with polynomial coefficients
type: Item
instance of: scholarly article
title: Meromorphic solutions of a differential equation with polynomial coefficients
zbMATH Open document ID: 1157.34063
DOI: 10.1007/BF03321665
author: Gary G. Gundersen
published in: Computational Methods and Function Theory
publication date: 2008-07-29T00:00:00Z
review text: The author considers a first order differential equation of the form \[ f'=P_0+P_1\cdot f+\dots+P_n\cdot f^n,\quad n\geq3 \tag{1} \] where each \(P_k\) is a polynomial, \(P_n\neq0\). Since \(n\geq3\), it follows from the Malmquist Theorem that any meromorphic solution of equation (1) must necessarily be a rational function. Furthermore, it was shown by G. G. Gundersen and I. Laine that (1) can possess at most a finite number of distinct meromorphic (rational) solutions. In the paper new estimates for the maximum number \(M\) of distinct meromorphic solutions and also for the maximum number \(L\) of linearly independent meromorphic solutions of the equation (1) are found. The estimate for \(M\) depends only on \(n\) and the number \(d\) of distinct zeros of \(P_n\), while the estimate for \(L\) depends only on \(d\).
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 5305515
zbMATH Keywords: differential equation
zbMATH Keywords: meromorphic solutions
zbMATH Keywords: rational solutions
zbMATH Keywords: linearly independent solutions
zbMATH Keywords: polynomial coefficients
reviewed by: A. Yu. Timofeev
MaRDI profile type: Publication
Wikidata QID: Q115390803
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description: scientific article; zbMATH DE number 5305515