Taylor Domination, Difference Equations, and Bautin Ideals

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.

This author is supported by ISF, Grants No. 639/09 and 779/13, and by the Minerva foundation.