Taylor Domination, Difference Equations, and Bautin Ideals

TL;DR

This paper compares three methods—Taylor domination, recurrence relations, and Bautin ideals—for analyzing the behavior of analytic functions through their Taylor coefficients, revealing their interconnections and raising new questions.

Contribution

It introduces and relates three approaches to study Taylor coefficients of analytic functions, highlighting their connections and proposing new research questions.

Findings

The three approaches are closely related.

Results connect Taylor domination, recurrences, and Bautin ideals.

Raises new questions about their interplay.

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 \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] 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 $\lambda$. \smallskip These three approaches turn out to be closely related. We present some results and questions in this direction.