Topological differential fields and dimension functions

Abstract

We construct a fibered dimension function in some topological differential fields.

Dimension in the realm of transseries

“…Differentially closed fields are d-bounded, as pointed out in [4]. Guzy and Point [7] (see also [3]) show that existentially closed ordered differential fields, and Scanlon's d-henselian valued differential fields with many constants (see [1,Chapter 8]) are d-bounded.…”

Strong Density of Definable Types and Closed Ordered Differential Fields

“…Instead of providing the original definition given in [GP12] of δ-dim, we will use a characterization in terms of the following closure operator, also proven to be equivalent in [GP12]. The following proposition characterizes the δ-dimension in terms of cl and gathers a second property that we will later need.…”

“…Alternatively, one can obtain the same dimension function proceeding as in [vdD89], as was done in [GP12] working in a broader differential setting (see [GP12, Proposition 2.13]). Here we will call such a dimension function the δ-dimension and denote it by δ-dim (in place of t-dim, the notation used in [GP12]). Throughout this section let U be a monster model of CODF and K ⊆ U be a small model.…”

“…Instead of providing the original definition given in [GP12] of δ-dim, we will use a characterization in terms of the following closure operator, also proven to be equivalent in [GP12]. For B ⊆ U and a ∈ U, say that a ∈ cl(B) if and only if there is a differential polynomial q ∈ Q B {x} \ {0} such that q(a) = 0.…”

Definable groups in topological differential fields

“…Using the same strategy as in [11], one defines a fibered dimension function on the definable subsets in models of T * c,D [16], proving that T * c,D has the equational boundedness property [16,Corollary 3.10]. For models of CODF , a fibered dimension function has been already introduced but using a cell-decomposition theorem [5].…”