Taylor Domination, Difference Equations, and Bautin Ideals
Conference paper
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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 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 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.
$$ |a_{k}|R^{k}\le C\ \max _{i=0,\dots ,N}\ |a_{i}|R^{i}, \ k \ge N+1. $$
$$ a_{k}=\sum _{j=1}^{d}c_{j}(k)\cdot a_{k-j}, \ k=d,d+1,\dots , $$
Keywords
Recurrence relations Bautin ideals Domination of initial Taylor coefficientsReferences
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