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 v E extending v to K ( X ) . After fixing an extension of v E to a fixed algebraic closure K ( X ) ‾ of K ( X ) , we show that the implicit constant field of the extension ( K ( X ) | K , v E ) 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 v E , provided that v has finite rank.