Gundersen, Gary G.; Laine, Ilpo Existence of meromorphic solutions of algebraic differential equations. (English) Zbl 0719.34010 Math. Scand. 67, No. 1, 35-55 (1990). We determine certain algebraic differential equations which do not possess any meromorphic solutions. We also determine the maximum number of distinct meromorphic solutions that certain other algebraic differential equations can possess. Several authors have given upper bounds for the number of distinct meromorphic solutions of the particular equation \((*)\quad f'=P_ 0(z)+P_ 1(z)f+...+P_ n(z)f^ n\) where \(n\geq 3\) and each \(P_ k(z)\) is a polynomial \((P_ n\not\equiv 0)\). We find new upper bounds for both the (i) maximum number of distinct meromorphic solutions of (*), and for the (ii) maximum number of linearly independent meromorphic solutions of (*). Examples and related results are given. Reviewer: Gary G.Gundersen Cited in 1 ReviewCited in 1 Document MSC: 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain Keywords:algebraic differential equations; meromorphic solutions; maximum number of linearly independent meromorphic solutions PDFBibTeX XMLCite \textit{G. G. Gundersen} and \textit{I. Laine}, Math. Scand. 67, No. 1, 35--55 (1990; Zbl 0719.34010) Full Text: DOI EuDML