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Bautin ideals and Taylor domination. (English) Zbl 1304.30004

Summary: We consider families of analytic functions with Taylor coefficients-polynomials in the parameter \(\lambda\):
\[ f_\lambda(z)=\sum_{k=0}^\infty a_k(\lambda)z^k,\quad a_k\in\mathbb C[\lambda]. \]
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 \(\lambda\),
\[ | a_k(\lambda)|\leq R^k(\lambda)\leq C\max_{i=0,\ldots,N}| a_i(\lambda)| R^i(\lambda). \]
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 Poincaré type linear recurrence relations.

MSC:

30B10 Power series (including lacunary series) in one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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