Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

Given two polynomials P, q we consider the following question: "how large can the index of the first non-zero moment mk=∫abP^kq be, assuming the sequence is not identically zero?" The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K≤2+deg q +3(deg P-1)². The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y'=py²+εqy³ for p, q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg q + 3deg² p. This is the first such bound depending solely on the degrees of the Abel equation.