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Taylor Domination, Difference Equations, and Bautin Ideals

  • Dmitry Batenkov
  • Yosef YomdinEmail author
Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 180)

Abstract

We compare three approaches to studying the behavior of an analytic function \(f(z)=\sum _{k=0}^\infty a_kz^k\) from its Taylor coefficients. The first is “Taylor domination” property for f(z) in the complex disk \(D_R\), which is an inequality of the form
$$ |a_{k}|R^{k}\le C\ \max _{i=0,\dots ,N}\ |a_{i}|R^{i}, \ k \ge N+1. $$
The second approach is based on a possibility to generate \(a_k\) via recurrence relations. Specifically, we consider linear non-stationary recurrences of the form
$$ a_{k}=\sum _{j=1}^{d}c_{j}(k)\cdot a_{k-j}, \ k=d,d+1,\dots , $$
with uniformly bounded coefficients. In the third approach we assume that \(a_k=a_k(\lambda )\) are polynomials in a finite-dimensional parameter \(\lambda \in {\mathbb C}^n.\) We study “Bautin ideals” \(I_k\) generated by \(a_{1}(\lambda ),\ldots ,a_{k}(\lambda )\) in the ring \({\mathbb C}[\lambda ]\) of polynomials in \(\lambda \). These three approaches turn out to be closely related. We present some results and questions in this direction.

Keywords

Recurrence relations Bautin ideals Domination of initial Taylor coefficients 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael

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