Groebner fan and embedded resolutions of ideals on toric varieties

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the "Groebner fan" of such an ideal is actually a polyhedral fan and that a sub-variety defined by a Newton non-degenerate ideal on a toric variety $X_\sigma$ admits a toric embedded resolution of singularities $Z\longrightarrow X_\sigma.$