scholarly journals Rank and Dimension in Difference-Differential Fields

2011 ◽  
Vol 52 (4) ◽  
pp. 403-414 ◽  
Author(s):  
Ronald F. Bustamante Medina
Keyword(s):  
Differential Fields

The model theory of ordered differential fields

1978 ◽  
Vol 43 (1) ◽  
pp. 82-91 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Meromorphic Functions ◽  
Analytic Solutions ◽  
Positive Element ◽  
Real Closed Field ◽  
Real Field ◽  
Closed Field ◽  
Ordered Field ◽  
Positive Elements ◽  
Odd Degree ◽  
Differential Fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

Valued Differential Fields

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Asymptotic Behavior ◽  
Residue Field ◽  
Field Extension ◽  
Field Extensions ◽  
Differential Field ◽  
Differential Fields ◽  
Dominant Part

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

scholarly journals Computing constraint sets for differential fields

2014 ◽  
Vol 407 ◽  
pp. 316-357 ◽  
Author(s):  
Russell Miller ◽  
Alexey Ovchinnikov ◽  
Dmitry Trushin
Keyword(s):  
Differential Fields ◽  
Constraint Sets
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