## Publicacions Matemàtiques

### Bautin ideals and Taylor domination

Y. Yomdin

#### 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

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

Mathematical Reviews number (MathSciNet)
MR3211848

Zentralblatt MATH identifier
1304.30004

#### Citation

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