It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.

中文翻译：

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关于Lyapunov和周期常数的注释

众所周知，对于一系列平面多项式矢量场，可以从弱焦点的起点或非退化中心的起点分叉的小幅度极限循环的数量由所谓的Lyapunov常数的结构控制。在系统的参数中。这些常数本质上是位移图为零时泰勒展开的奇数项的系数。尽管许多作者认为该图的偶数项的系数属于由先前的奇数项生成的理想值，但我们在文献中找不到证据。在本文中，我们基于一维解析可逆取向微分形本身的一般性质，提供了对此事实的简单证明。我们还证明了周期常数的相似结果。