Bautin ideals and Taylor domination

Abstract

We consider families of analytic functions with Taylor coefficients\guio{polynomials} in the parameter $\lambda$: $f_\lambda(z)=\sum_{k=0}^\infty a_k(\lambda) z^k$, $a_k \in {\mathbb C}[\lambda]$. Let $R(\lambda)$ be the radius of convergence of $f_\lambda$. The "Taylor domination'' property for this family is the inequality of the following form: for certain fixed~$N$ and $C$ and for each $k\geq N+1$ and $\lambda,

$|a_{k}(\lambda)|R^{k}(\lambda)\leq C \max_{i=0,\dotsc,N} |a_{i}(\lambda)|R^{i}(\lambda).$

Taylor domination property implies a uniform in $\lambda$ bound on the number of zeroes of~$f_\lambda$. In this paper we discuss some known and new results providing Taylor domination (usually, in a smaller disk) via the Bautin approach. In particular, we give new conditions on $f_\lambda$ which imply Taylor domination in the full disk of convergence. We discuss Taylor domination property also for the generating functions of the Poincar\'e type linear recurrence relations.