Idealistic exponents: Tangent cone, ridge, characteristic polyhedra

We study Hironaka’s idealistic exponents over $ \operatorname{Spec} ( $\mathbb{Z}$ ) $. We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.