Frank, Günter; Wang, Yuefei On the meromorphic solutions of algebraic differential equations. (English) Zbl 0915.34007 Analysis, München 18, No. 1, 49-54 (1998). Estimates for the growth of meromorphic solutions to algebraic differential equations of arbitrary order of the form (1) are obtained.The authors consider algebraic differential equations of the form \[ P_0(z,w)\Biggl({dw\over dz}\Biggr)^n= \sum^m_{j= 1} P_j(z,w)D_j[w],\tag{1} \] whith \(n,m\in \mathbb{N}\); \(P_j(z,w)\) are polynomials in \(z\) and \(w\) and \(P_0(z,w)\not\equiv 0\); \(D_j[w]\) are differential monomials in \(w\) of the form \[ D_j[a]= \Biggl({dw\over dz}\Biggr)^{j_1} \Biggl({d^2w\over dz^2}\Biggr)^{j_2}\cdots \Biggl({d^lw\over dz^l}\Biggr)^{j_l}, \] with \(j_1,j_2,\dots, j_l\in\mathbb{N}\cup \{0\}\), for \(j= 0,1,\dots, m\). The authors call \(\nu_j= j_1+ 2j_2+\cdots+ lj_l\) the weight of \(D_j\). The weight of \[ P[w](z)= \sum^m_{j= 1} P_j(z,w)D_j[w] \] is defined by \(\nu(P)= \max_{1\leq j\leq m}\{\nu_j\}\). Cited in 5 Documents MSC: 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain 34C11 Growth and boundedness of solutions to ordinary differential equations 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory Keywords:estimates; growth; meromorphic solutions; algebraic differential equations; arbitrary order PDFBibTeX XMLCite \textit{G. Frank} and \textit{Y. Wang}, Analysis, München 18, No. 1, 49--54 (1998; Zbl 0915.34007) Full Text: DOI