  Given an irreducible hypersurface singularity of dimension $d$ (defined by a
polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space
defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether
the singularity is quasi-ordinary with respect to the projection. The
construction uses a weighted version of Hironaka's characteristic polyhedron
and successive embeddings of the singularity in affine spaces of higher
dimensions. When $ f $ is quasi-ordinary, our invariant determines the
semigroup of the singularity and hence it encodes the embedded topology of the
singularity $ \{ f = 0 \} $ in a neighbourhood of the origin when $ K =
\mathbb{C};$ moreover, the construction yields the approximate roots, giving a
new point of view on this subject.
