Algebraic solutions of algebraic differential equations
type: Item
instance of: scholarly article
title: Algebraic solutions of algebraic differential equations
zbMATH Open document ID: 1071.30026
published in: Applied Mathematics. Series B (English Edition)
publication date: 2005-05-06T00:00:00Z
review text: The author gives a condition under which a second-order algebraic differential equation has an algebraic solution. Let \(a_0\dots, a_p\), \(b_0,\dots, q\) be nonzero entire functions of one variable such that they have a finite number of poles and without common zero, and consider the following equation: \[ (w'')^n= \Biggl(\sum^p_{i=0} a_i(z) w^i\Biggr)\Biggl/\Biggl(\sum^q_{i=0} b_i(z) w^i\Biggr). \] Suppose that the above equation has at least one nonconstant \(\nu\)-valued algebroid solution \(w\) on \(\mathbb{C}\). Then he proves that that \(w\) is algebraic if all \(a_i\), \(b_j\) are polynomials and \(p< n+ q\).
reviewed by: Yoshihiro Aihara
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 2165846
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: algebroid function
zbMATH Keywords: algebraic differential equation
author: Lingyun Gao
MaRDI profile type: Publication
full work available at URL: s11766-005-0035-3
OpenAlex ID: W2332577534
cites work: Q3317367
cites work: On algebroid solutions of some binomial differential equations in the complex plane
cites work: Admissible solutions of ordinary differential equations
cites work: On some algebraic differential equations with admissible algebroid solutions
cites work: Some results on admissible algebroid solutions of complex differential equations
cites work: Q4872312
cites work: LOGARITHMIC DERIVATIVES OF MEROMORPHIC OR ALGEBROID SOLUTIONS OF SOME HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS
cites work: On a theorem of gol'dberg concerning meromorphic solutions of algebraic differential equations
DOI: 10.1007/S11766-005-0035-3
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description: scientific article; zbMATH DE number 2165846