On Dimensions, Standard Part Maps, and p-Adically Closed Fields (original) (raw)

The aim of this paper is to study the dimensions and standard part maps between the field of p-adic numbers Q p and its elementary extension K in the language of rings L r. We show that for any K-definable set X ⊆ K m , dim K (X) ≥ dim Qp (X ∩ Q m p). Let V ⊆ K be convex hull of K over Q p , and st : V → Q p be the standard part map. We show that for any K-definable function f : K m → K, there is definable subset D ⊆ Q m p such that Q m p \D has no interior, and for all x ∈ D, either f (x) ∈ V and st(f (st −1 (x))) is constant, or f (st −1 (x)) ∩ V = ∅. We also prove that dim K (X) ≥ dim Qp (st(X ∩ V m)) for every definable X ⊆ K m .