Diophantine approximations and toric deformations

We propose a reformulation of the Faltings-Wüstholz nonlinear version of Schmidtʹs subspace theorem with the help of toric deformations and Chow polytopes. Moreover, we show that the arithmetic Bezout theorem in Arakelov geometry can be used to obtain a Bezout theorem for Mumfordʹs degree of contact. This is a birational invariant often considered in geometric invariant theory (GIT). The originality of this last result relies on the interpretation of GIT as a degeneration of Arakelov geometry. This should enable us to transfer all known results of Arakelov geometry into GIT.