We build two toric embedded resolutions procedures of a reduced quasiordinary hypersurface singularity (S, 0) of dimension d . The first one provides an embedded resolution as hypersurface of (C d+1 , 0) as a composition of toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection (S, 0) → (C d , 0) . This gives a positive answer to a a question of Lipman (see [L5]). The second method applies to the analytically irreducible case; if g ≥ 1 denotes the number of characteristic monomials we re-embed the germ (S, 0) in the affine space (C d+g , 0) by using certain approximate roots of a suitable Weierstrass polynomial defining the embedding (S, 0) ⊂ (C d+1 , 0) . We build a toric morphism which is a simultaneous toric embedded resolution of the irreducible germ (S, 0) ⊂ (C d+g , 0) , and of an affine toric variety Z Γ obtained from (S, 0) ⊂ (C d+g , 0) by specialization and defined by a rank d semigroup Γ generalizing the classical semigroup of a plane branch. Finally we compare both resolutions and we prove that the first one is the restriction of the second to a smooth (d + 1) -variety containing the strict transform of S .

Abstract. The geometric motivic Poincaré series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic jacobian ideals, associated to the modules of differential forms with logarithmic poles outside the torus of the toric variety.

An analytically irreducible hypersurface germ (S, 0) ⊂ (C d+1 , 0) is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial f ∈ C{X}[Y ] of a fractional power series in the variables X = (X 1 , . . . , X d ) which has characteristic monomials, generalizing the classical Newton-Puiseux characteristic exponents of the plane-branch case (d = 1). We prove that the set of vertices of Newton polyhedra of resultants of f and h with respect to the indeterminate Y , for those polynomials h which are not divisible by f , is a semigroup of rank d, generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial f are irreducible quasiordinary polynomials and that, together with the coordinates X 1 , . . . , X d , provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of (S, 0) are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ (S, 0) as characterized by the work of Gau and Lipman.A quasi-ordinary hypersurface germ can be defined by an equation. The Jung-Abhyankar theorem implies that the roots of quasi-ordinary polynomials, called quasi-ordinary branches, are fractional power series in the ring C{X 1/n } for some positive integer n (see [J, A1]). Since the difference ζ − ζ of any two roots of f divides the discriminant, it must be of the form X λ , where ∈ C{X 1/n } is a unit and λ is a dtuple of non-negative rationals. The monomials X λ so obtained are called characteristic monomials.Lipman builds an inversion lemma that associates to any quasi-ordinary branch ζ a normalized quasi-ordinary branch parametrizing the same germ whose characteristic monomials are obtained from those of ζ by an inversion formulae similar to that of the plane-curve case (see [L1] and the appendix of [Gau]). Being normalized is a technical condition that in the plane-curve case means that the projection (X, Y ) → X is transversal. In the two-dimensional case, Lipman proved that the characteristic monomials of a normalized quasi-ordinary branch are an analytical invariant of the surface (see [L1, L3]). Luengo gives another proof of this result (see [Lu]). Lipman remarked, using general results of Zariski on saturation of local rings, that the characteristic monomials of a normalized quasi-ordinary branch determine the topological type of the germ it parametrizes (see [L4] and also [Oh] for another proof); Gau proved the converse: these monomials define a complete invariant of the embedded topological type of the germ. Gau's proof involves some results of Lipman on topological invariants of quasi-ordinary singularities: the description of the local divisor class group in terms of the characteristic monomials (see [Gau,L4]).When f is an irreducible quasi-ordinary polynomial, we can generalize some of the properties of the intersection multiplicity of plane-curve...