The Hilbert 16-th problem and an estimate for cyclicity of an elementary   polycycle
  Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of limit cycles (isolated periodic solutions), denoted by LC(\eps), over the parameter \eps, i.e. max_{\eps \in B^k} LC(\eps) <= K < \infty for some K.   The HA problem can be reduced to so-called Local Hilbert-Arnold (LHA) problem. Suppose that a generic k-parameter family {\dot x=v(x,\eps)}_{\eps \in B^k}, x\in S^2 for some parameter \eps^*\in B^k has a polycycle (separatrix polygon) gamma consisting of equilibrium points as vertices and connecting separatrices as sides. LHA problem is to estimate B(k)--- the maximal number of limit cycles that can be born in a neighbourhood of gamma for a field \dot x=v(x,\eps), where \eps is close to \eps^*. 
Vadim Kaloshin (American institute of Mathematics)