Newton non-degenerate $\mu$-constant deformations admit simultaneous embedded resolutions

Abstract

Let $\mathbb{C}^{n+1}_o$ denote the germ of $\mathbb{C}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $\mathbb{C}^{n+1}_o$ and $W$ a deformation of $V$ over $\mathbb{C}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we will prove that $W$ is a $\mu$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(W(\mathbb{C}_o)_m)_{red} \rightarrow \mathbb{C}_{o}$ is flat, where $W(\mathbb{C}_o)_m$ is the relative space of $m$-jets. On the way tothe proof of our main result, we give a complete answer to a question ofArnold on the monotonicity of Newton numbers in the case of convenientNewton polyhedra.