  We initiate a classification of complex polynomials f of degree d having the
top Betti number of the general fibre close to the maximum. We find a range in
which the polynomial must have isolated singularities and another range where
it may have at most a line singularity of Morse transversal type, besides
controlled singularities at infinity. Our method uses deformations into
particular pencils with non-isolated singularities.
