Abel微分方程零解循环性的一致上界
Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation
摘要：
给定两个多项式$P,q$，我们考虑以下问题：“假设序列不全为零，第一非零矩$\tilde{m}_k=\int_a^b P^k q$的指数可以有多大？".这个问题的答案$K$被称为时刻Bautin指数，我们提供了第一个通用的上限：$K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$。证明是基于定性分析的线性常微分方程，适用于柯西型积分的某些代数函数。矩Bautin指数在研究多项式Abel方程$y'=py^2+\varepsilon qy^3$（$p,q$多项式）和$\varepsilon \ll 1$的周期解分支中起着重要的作用.特别是，我们的结果意味着，对于$p$满足一个众所周知的一般条件，零解附近的周期解的数量不超过$5+\mathrm{deg} q+3\mathrm{deg}^2 p$。这是第一个这样的界限完全取决于阿贝尔方程的程度。
英文摘要：
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.
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