Upper bounds of topology of complex polynomials in two variables

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two distinct critical values. We prove quantitative versions of this statement. Supposing $H$ appropriately normalized (by affine coordinate changes in the image and in the source) we prove upper bounds for the following quantities: - the sum of the coefficients of the lower terms; - the minimal size of a bidisc containing all the nontrivial topology of a given level curve $S_t=\{ H=t\}$; - the minimal lengths of representatives of cycles in $H_1(S_t,\zz)$ vanishing along appropriate paths from $t$ to the critical values of $H$; - the intersection indices of the latter cycles. All these results (expect for the latter bound) are used in my joint work with Yu.S.Ilyashenko "Restricted version of the Hilbert 16-th problem" (available on the arxiv). In the latter paper we obtain an explicit upper bound of the number of zeros for a wide class of Abelian integrals.