Open Access
2014 Bautin ideals and Taylor domination
Y. Yomdin
Publ. Mat. 58(S1): 529-541 (2014).


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.


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Y. Yomdin. "Bautin ideals and Taylor domination." Publ. Mat. 58 (S1) 529 - 541, 2014.


Published: 2014
First available in Project Euclid: 19 May 2014

zbMATH: 1304.30004
MathSciNet: MR3211848

Primary: 30B10 , 34C05 , 34C25

Keywords: Bautin ideals , Poincarç-type recurrence , Taylor domination , Turan Lemma

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. S1 • 2014
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