Criteria to bound the number of critical periods (Q1011474)

The present paper is devoted to a criterion that allows to bound the number of critical periods of a center (degenerate or nondegenerate) of Hamiltonian systems \(\dot{x}=y,\) \(\dot{y}=V'(x),\) where \(V(x)=x^{2m}/{2m}+o(x^{2m})\) is an analytic function in a neighbourhood of \(x=0\). The authors first apply their criterion to already known results to show that it avoids long computations. Then it is proved that the problem of bounding the number of critical periods reduces to a purely algebraic problem, namely, to count zeros of a polynomial (sometimes depending on a parameter).