## Publicacions Matemàtiques

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 May 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1400505247

**Mathematical Reviews number (MathSciNet)**

MR3211848

**Zentralblatt MATH identifier**

1304.30004

**Subjects**

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

**Keywords**

Bautin ideals Taylor domination Turan Lemma Poincarç-type recurrence

#### Citation

Yomdin, Y. Bautin ideals and Taylor domination. Publ. Mat. EXTRA (2014), 529--541. https://projecteuclid.org/euclid.pm/1400505247