Betti bounds of polynomials

We initiate a classification of polynomials $f\colon\mathbb C^n\to\mathbb C$ 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 one line singularity of general Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularities.