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 m˜k=∫abPkq 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.