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$\mu$-constant deformations admit simultaneous embedded resolutions
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 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 
$(\operatorname {W( {\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where 
$\operatorname {W( {\mathbb {C}}_{o})}_{m}$ is the relative space of 
$m$-jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra.
