Weak Infinitesimal Hilbert's 16th Problem

The following weak infinitesimal Hilbert's 16th problem is solved. Given a real polynomial $H$ in two variables, denote by $M(H,m)$ the maximal number possessing the following property: for any generic set $\{\gamma _i\}$ of at most $M(H,m)$ compact connected components of the level lines $H=c_i$ of the polynomial $H$, there exists a form $\omega =P\,dx+Q\,dy$ with polynomials $P$ and $Q$ of degrees no greater than $m$ such that the integral $\int _{H=c}\omega$ has nonmultiple zeros on the connected components $\{\gamma _i\}$. An upper bound for the number $M(H,m)$ in terms of the degree $n$ of the polynomial $H$ is found; this estimate is sharp for almost every polynomial $H$ of degree $n$. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert's 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree $n$ have if it is close to a Hamiltonian vector field?