Newton polygons and families of polynomials

Abstract

We consider a continuous family $(f_s)$, $s\in [0,1]$ of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant (or equivalently the sum of the affine Milnor number and the Milnor number at infinity $\mu (s)+\lambda (s)$ is constant). We firstly prove that the set of critical values at infinity depends continuously on $s$, and secondly that the degree of the $f_s$ is constant (up to an algebraic automorphism of $\Cc^2$).

12 pages, 8 figures. Final version, to appear in Manuscripta Mathematica