Betti Bounds of Polynomials

Abstract

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.