In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain the lower bounds for the number of critical periods for these systems. For polynomial potential systems of degree $n$, we provide a lower bound of $n-2$ for the number of critical periods, and for polynomial systems of degree $n$, we acquire a lower bound of $n^2/2+n-5/2$ when $n$ is odd and $n^2/2-2$ when $n$ is even for the number of critical periods. To the best of our knowledge, these lower bounds are new, moreover the latter one is twice the existing results up to the dominant term.

中文翻译：

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平面多项式系统临界周期数的新下限

在本文中，我们构造了两类以原点为中心的平面多项式哈密顿系统，并获得了这些系统的临界周期数的下界。对于 n$ 次多项式势系统，我们提供了 $n-2$ 的临界周期数下界，对于 n$ 次多项式系统，我们获得了 $n^2/2 的下限+n-5/2$，当 $n$ 为奇数时，$n^2/2-2$ 当 $n$ 为偶数时，关键时期的数量。据我们所知，这些下限是新的，而且后一个是现有结果的两倍，直到主导项。