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

Authors
Type
Preprint
Publication Date
Apr 09, 2015
Submission Date
Apr 09, 2015
Identifiers
arXiv ID: 1504.02208
Source
arXiv
Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ 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: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. 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 $y'=py^2+\varepsilon qy^3$ for $p,q$ polynomials and $\varepsilon \ll 1$. 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 $5+\mathrm{deg} q+3\mathrm{deg}^2 p$. This is the first such bound depending solely on the degrees of the Abel equation.