On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

Abstract

Given a valued field (K, v) and a pseudo monotone sequence E in (K, v), one has an induced valuation vE extending v to K(X). After fixing an extension of vE to a fixed algebraic closure K(X) of K(X), we show that the implicit constant field of the extension (K(X)|K, vE) is simply the henselization of (K, v). We consider the question: given a value transcendental extension w of v to K(X) and a pseudo monotone sequence E in (K, v), under which precise conditions are w induced by E? The dual nature of pseudo convergent sequences of algebraic type and pseudo divergent sequences is also explored. Further, we provide a complete description of the various possibilities of the rank of the valuation vE , provided that v has finite rank.