Algebraic families of analytic functions. I
type: Item
instance of: scholarly article
title: Algebraic families of analytic functions. I
zbMATH Open document ID: 0886.34005
published in: Journal of Differential Equations
publication date: 1998-05-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 1025920
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: algebraic functions
zbMATH Keywords: polynomial differential equations
zbMATH Keywords: Bautin ideal
author: M. Briskin
author: Yosef Yomdin
reviewed by: Günter Czichowski
MaRDI profile type: Publication
full work available at URL: jdeq.1996.3250
OpenAlex ID: W2046807088
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DOI: 10.1006/JDEQ.1996.3250
description: scientific article; zbMATH DE number 1025920