Definable henselian valuation rings

Ora Inizio: 
Ora Fine: 
Alexander Prestel
Universität Konstanz, Germania

A valuation ring O of a field K is called "henselian" if every polynomial in one variable with coefficients from O that has a simple zero in the quotient of O w.r.t. its maximal ideal, has already a zero in K. Such henselian fields are for example  local fields and all generalized power series fields. In all these examples the valuation ring O remains fixed under all field automorphisms from K. The reason for this is that O can be defined by a first order formula in the language of rings.
We shall discuss the question when O is definable this way, and if so, how complicate the a defining formula may be. Using model theoretic arguments, in most cases we shall only obtain the existence of such a formula (even though it might be quite simple) without knowing it explicitly.