Approximation Theorems

Abstract

We embed the important work of Gräter on approximation theorems in the book. Approximation theorems are a well-known and important topic in classical valuation theory of fields. The question is to decide for given valuations v ₁, …, v _n of a field, elements a ₁, …, a _n in the field and α ₁, …, α _n in the value groups whether there is an element x in the field such that

$$\displaystyle{v_{i}(x - a_{i}) \geq \alpha _{i}\mbox{ resp. }v_{i}(x - a_{i}) =\alpha _{i}}$$

for all i; i.e. if the elements a _i can be approximated by some x up to a certain degree. Gräter elaborated various approximation theorems in our general setting of R-Prüfer rings and has found deep connections, to be reflected below.