  Let $X$ be a planar smooth vector field with a polycycle $\Gamma^n$ with $n$
sides and all its corners, that are at most $n$ singularities, being hyperbolic
saddles. In this paper we study the cyclicity of $\Gamma^n$ in terms of the
hyperbolicity ratios of these saddles, giving explicit conditions that ensure
that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results
and also provides a more accurate proof of the known ones because we rely on
some recent powerful works that study in more detail the regularity with
respect to initial conditions and parameters of the Dulac map of hyperbolic
saddles for families of vector fields. We also prove that when $X$ is
polynomial there is a polynomial perturbation (in general with degree much
higher that the one of $X$) that attains each of the obtained lower bounds for
the cyclicities. Finally, we also study some related inverse problems and
provide concrete examples of applications in the polynomial world.
