We study the weak Hilbert problem for the Hamiltonian family [Formula: see text] [Formula: see text] under arbitrary polynomial perturbations of degree [Formula: see text], where [Formula: see text]. Using the first-order averaging method in generalized polar coordinates, we prove that the maximum number of limit cycles that can bifurcate from the periodic orbits of the unperturbed center is at least [Formula: see text].
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Anacona et al. (2026) studied this question.
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