DEFORMATIONS WITH CONSTANT MILNOR NUMBER AND MULTIPLICITY OF COMPLEX HYPERSURFACES

Abstract

We investigate the constancy of the Milnor number of one parameter deformations of holomorphic germs of functions $f:(\C^n,0) \to (\C,0)$ with isolated singularity, in terms of some Newton polyhedra associated to such germs.

When the Jacobian ideals $J(\hspace*{1.8pt}f_t) = \left\langle {\partial f_t}/{\partial x_{1}} \ldots ,{\partial f_t}/{\partial x_{n}}\right\rangle $ of a deformation $f_t(x) = f(x)+ \sum_{s=1}^{\ell}\delta_s(t)g_s(x)$ are non-degenerate on some fixed Newton polyhedron $\Gamma_+$, we show that this family has constant Milnor number for small values of $t$, if and only if all germs $g_s$ have non-decreasing $\Gamma$-order with respect to $f$. As a consequence of these results we give a positive answer to Zariski's question for Milnor constant families satisfying a non-degeneracy condition on the Jacobian ideals.