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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.

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