On minimal pairs and residually transcendental extensions of valuations

Abstract

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure λ̅ of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to λ̅ is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.