Newton non-degenerate -constant deformations admit simultaneous embedded resolutions

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