  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.
