The complex case of Schmidt's going-down Theorem

TL;DR

This paper revisits Schmidt's going-down Theorem, clarifies its proof in the complex case for non-totally real fields, and extends related inequalities to arbitrary number fields using multilinear algebra.

Contribution

It provides a detailed proof of Schmidt's going-down Theorem in the complex case and generalizes Laurent's inequalities to arbitrary number fields.

Findings

Proof of Schmidt's complex case clarified and completed.

Laurent's inequalities extended to arbitrary number fields.

Use of multilinear algebra and wedge products in the reformulation.

Abstract

In 1967, Schmidt wrote a seminal paper [10] on heights of subspaces of R n or C n defined over a number field K, and diophantine approximation problems. The going-down Theorem -- one of the main theorems he proved in his paper -- remains valid in two cases depending on whether the embedding of K in the complex field C is a real or a complex non-real embedding. For the latter, and more generally as soon as K is not totally real, at some point of the proof, the arguments in [10] do not exactly work as announced. In this note, Schmidt's ideas are worked out in details and his proof of the complex case is presented, solving the aforementioned problem. Some definitions of Schmidt are reformulated in terms of multilinear algebra and wedge product, following the approaches of Laurent [5], Bugeaud and Laurent [1] and Roy [7], [8]. In [5] Laurent introduces in the case K = Q a family of exponents and he gives a series of inequalities relating them. In Section 5 these exponents are defined for an arbitrary number field K. Using the going-up and the going-down Theorems Laurent's inequalities are generalized to this setting.