Bautin Ideals and Taylor Domination

We consider families of analytic functions with Taylor coefficients-polynomials in the parameter λ: fλ(z) = ∑∞ k=0 ak(λ)z k, 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(λ) ≤ C max i=0,...,N |ai(λ)|R(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. 2010 Mathematics Subject Classification: 34C05, 34C25, 30B10.