Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

We will discuss the following results, from joint work with R. Miller and C. Hall.

Let p,l,q be rational prime numbers, c a positive integer. Let F_l(t) be a rational function field over a finite field of characteristic l>0. Let v be a discrete valuation on F_l(t), and let U be the completion of F_l(t) under v. Let L be an algebraic extension of Q_p or U such that for any finite subfield K of L containing Q_p (resp. U) we have that ord_q([K:Q_p])<c (resp. ord_q([K:U])<c). Then we will call L a q-bounded extension of a local field.  

Now let F be a function field over a field H of constants embeddable into a q-bounded extension of a local field, where in the case of positive characteristic we assume that H contains F_l(t).  Let m be any function field valuation on F (i.e. m is trivial on H).  Let w in F\H.  Let Q^{alg} be the algebraic closure of Q or F_l(t) in H, and let F^{alg}_w be the algebraic closure of Q^{alg}(w) in F.

1.  If H is henselian or is algebraic over a global field, then the valuation ring of m is existentially definable over F.

2.  For any H as above and any m non-trivial on F^{alg}_w, there exists a subset V_m of F, Diophantine over F and  such that the following conditions are satisfied:

--- If x in V_m, then ord_{m}(x) >= 0.
--- If x in F^{alg}_w, and ord_m(x) >= 0, then x is in V_m.
 
3.  For any H as above, H10 is undecidable over F.