Bautin ideals and Taylor domination

• Autores: Yosef Yomdin
• Localización: Publicacions matematiques, ISSN 0214-1493, Nº. Extra 0 (Proceedings of the Conference “New Trends in Dynamical Systems” held in Salou (Tarragona), Spain, 1-), 2014, págs. 529-541
• Idioma: inglés
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    This research was supported by the ISF, grant no. 639/09 and by the Minerva foundation.

  • English

    We consider families of analytic functions with Taylor coefficients-polynomials in the parameter λ: fλ(z) = P∞k=0 ak(λ)zk, ak ∈ C[λ]. Let R(λ) be the radius of convergence of fλ. The “Taylor domination” property for this family is the inequality of the following form: for certain fixed N and C and for each k ≥ N + 1 and λ, |ak(λ)|R k (λ) ≤ C max i=0,...,N |ai(λ)|R i (λ). Taylor domination property implies a uniform in λ bound on the number of zeroes of fλ. 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 fλ 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.