Home /  DDC - Valuation Theory: Existential definability of valuations over function fields with constant fields embeddable into q-bounded algebraic extensions of local fields and H10 over these fields

Seminar

DDC - Valuation Theory: Existential definability of valuations over function fields with constant fields embeddable into q-bounded algebraic extensions of local fields and H10 over these fields September 23, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program:
Location: SLMath: Online/Virtual
Speaker(s) Alexandra Shlapentokh (East Carolina University)
Description

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

Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Existential Definability Of Valuations Over Function Fields With Constant Fields Embeddable Into Q-Bounded Algebraic Extensions Of Local Fields And H10 Over These Fields

Abstract/Media

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.

 

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Existential Definability Of Valuations Over Function Fields With Constant Fields Embeddable Into Q-Bounded Algebraic Extensions Of Local Fields And H10 Over These Fields

H.264 Video 25140_28638_8514_Existential_Definability_of_Valuations_Over_Function_Fields_with_Constant_Fields_Embeddable_into_Q_Bounded_Algebraic_Extensions_of_Local_Fields_and_H10_Over_These_Fields.mp4