Taylor Domination, Difference Equations, and Bautin Ideals

We compare three approaches to studying the behavior of an analytic function $f(z)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{z}^k$ 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 $$\mathrm{\mid }{a}_k\mathrm{\mid }{R}^k\le C\underset{i=0,\dots ,N}{\max }\mathrm{\mid }{a}_i\mathrm{\mid }{R}^i,k\ge N+1.$$ 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 $\lambda \in {\mathbb{C}}^n\mathrm{.}$ 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.