Superstable differential fields

1992 ◽  
Vol 57 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Pillay ◽  
Ž. Sokolović
Keyword(s):  
Differential Forms ◽  
Quantifier Elimination ◽  
Normal Extension ◽  
Additional Structure ◽  
Unique Type ◽  
Generic Type ◽  
Differential Field ◽  
Closed Fields ◽  
Invariant Differential ◽  
Differential Fields

In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field K with quantifier elimination is ω-stable. (Any quantifier free type over K defines a unique type over K in the sense of dc(k), the differential closure of K, and as we mention below the theory of differentially closed fields is ω-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension L of an algebraically closed differential field K is generated over K by an element η of some algebraic group G defined over CK, the constants of K, where η satisfies some specific differential equations over K related to invariant differential forms on G (η is “G-primitive” over K), and (ii) the fact that a superstable field has a unique generic type which is semiregular.

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).

Differential Polynomials

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Characteristic Zero ◽  
Natural Setting ◽  
Differential Polynomials ◽  
Differential Field ◽  
Closed Fields ◽  
Basic Facts ◽  
Differential Fields

This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their extensions. It then considers various decompositions of differential polynomials in their natural setting, along with valued differential fields and the property of continuity of the derivation with respect to the valuation topology. It also discusses the gaussian extension of the valuation to the ring of differential polynomials and concludes with some basic results on simple differential rings and differentially closed fields. In contrast to the corresponding notions for fields, the chapter shows that differential fields always have proper d-algebraic extensions, and the differential closure of a differential field K is not always minimal over K.

scholarly journals Asymptotic Differential Algebra and Model Theory of Transseries

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Algebraic Closure ◽  
Differential Algebra ◽  
Quantifier Elimination ◽  
Point Of View ◽  
Algebraic Point ◽  
Differential Field ◽  
Algebraic Setting ◽  
Differential Fields

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

scholarly journals A construction of the general relativistic Boltzmann equation

1984 ◽  
Vol 7 (3) ◽  
pp. 591-597 ◽  
Author(s):  
P. Dolan ◽  
A. C. Zenios
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Kinetic Equation ◽  
Simple Model ◽  
State Space ◽  
Differential Forms ◽  
Relativistic Boltzmann Equation ◽  
Dimensional State Space ◽  
General Relativistic ◽  
Invariant Differential

Our work depends essentially on the notion of a one-particle seven-dimensional state-space. In constructing a general relativistic theory we assume that all measurable quantities arise from invariant differential forms. In this paper, we study only the case when instantaneous, binary, elastic collisions occur between the particles of the gas. With a simple model for colliding particles and their collisions, we derive the kinetic equation, which gives the change of the distribution function along flows in state-space.

Duality on the quantum space(3) with two parameters

Open Physics ◽  
2012 ◽  
Vol 10 (5) ◽  
Author(s):  
Muttalip Özavşar ◽  
Gürsel Yeşilot
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Differential Forms ◽  
Quantum Space ◽  
Dual Algebra ◽  
Commutation Relations ◽  
Dual Hopf Algebra ◽  
Invariant Differential ◽  
Leibniz Rules ◽  
Two Parameters

AbstractIn this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

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.

scholarly journals Moving Frames for Lie Pseudo–Groups

2008 ◽  
Vol 60 (6) ◽  
pp. 1336-1386 ◽  
Author(s):  
Peter J. Olver ◽  
Juha Pohjanpelto
Keyword(s):  
Recurrence Relations ◽  
Differential Forms ◽  
Group Actions ◽  
Differential Invariants ◽  
Moving Frames ◽  
Constructive Theory ◽  
Invariant Differential

AbstractWe propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. Themoving frame provides an effectivemeans for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

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