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Algebraic families of analytic functions. I. (English) Zbl 0886.34005
There are investigated formal power series \[ y(x)= \sum_{n=1}^\infty a_k(\lambda) x^n \] as algebraic functions satisfying \(y=p_0(x)+ p_1(x)y+\dots+ p_d(x)y^d\) or in an analogous way as solutions of polynomial differential equations \[ y'= p_0(x)+ p_1(x)y+\dots+ p_d(x)y^d, \] \(\lambda= (\lambda_{ij})\in \mathbb{C}^{(d+1)}\) is a tupel of parameters.
By recurrence formulae one gets certain estimations for the polynomial coefficients \(a_k(\lambda)\).
This leads to the membership of the regarded algebraic functions to so-called Bernstein classes and to statements on the numbers of zeros.
Furthermore these constructions are in an algebraic context related to so-called \(A_0\)-series consisting of all series \(f_\lambda(x)= \sum_{n=0}^\infty a_n(x)x^n\), where the degree of \(a_j(\lambda)\) increases linearly in \(j\) and the norm \(|a_j(\lambda)|\) increases exponentially. The ideal generated from all \(a_k(\lambda)\) is the so-called Bautin ideal of \(f_\lambda\), and properties of the above series \(y(x)\) are related to this ideal. The last part of the paper concerns Bautin ideals of solutions of higher order linear differential equations.

MSC:
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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