Taylor Domination, Difference Equations, and Bautin Ideals

Abstract

We compare three approaches to studying the behavior of an analytic function $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ 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 $a_1(\lambda ),\dots ,a_k(\lambda )$ 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.

arXiv admin note: substantial text overlap with arXiv:1301.6033

References in corpus (1)

• Moment vanishing of piecewise solutions of linear ODEs