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On the meromorphic solutions of algebraic differential equations. (English) Zbl 0915.34007

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 \(\nu(P)= \max_{1\leq j\leq m}\{\nu_j\}\).

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
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