| Contents |
| We deal with isochronous centers of the so-called quadratic-like Hamiltonian systems, i.e., planar differential systems of the form
\[
\left\{
\begin{array}{l}
\dot x=-H_y(x,y), \\ \dot y=H_x(x,y),
\end{array}
\right.
\] where the Hamiltonian function is $H(x,y)=A(x)+B(x)y+C(x)y^2.$ A necessary and sufficient condition for isochronicity in case that $A$, $B$ and $C$ are analytic was given in [A. Cima, F. Ma\~{n}osas and J. Villadelprat, "Isochronicity for several classes of Hamiltonian systems", J. Differential Equations, 157 (1999) 373-413]. An involution $\sigma$ inherited from the geometry of the periodic orbits surrounding the center plays a key role in this condition. In the present talk we will explain some recent results on the case that $A$, $B$ and $C$ are polynomials. Our results show that in order to study the polynomial isochrones one can reduce, essentially, to the case in which the involution $\sigma$ is a M\"{o}bius transformation. |
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