## Abstract

Given two polynomials P, q we consider the following question: "how large can the index of the first non-zero moment mk=∫abP^{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≤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'=py^{2}+εqy^{3} 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 + 3deg^{2} p. This is the first such bound depending solely on the degrees of the Abel equation.

Original language | English |
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Pages (from-to) | 5769-5781 |

Number of pages | 13 |

Journal | Journal of Differential Equations |

Volume | 259 |

Issue number | 11 |

DOIs | |

State | Published - 5 Dec 2015 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics