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Bautin ideals and Taylor domination

Y. Yomdin

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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.

Article information

Publ. Mat., Volume EXTRA (2014), 529-541.

First available in Project Euclid: 19 May 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C05: Location of integral curves, singular points, limit cycles 34C25: Periodic solutions 30B10: Power series (including lacunary series)

Bautin ideals Taylor domination Turan Lemma Poincarç-type recurrence


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

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