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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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