In
mathematics, a
differential field
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
''K'' is differentially closed if every finite system of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s with a solution in some differential field extending ''K'' already has a solution in ''K''. This concept was introduced by . Differentially closed fields are the analogues
for differential equations of algebraically closed fields for polynomial equations.
The theory of differentially closed fields
We recall that a
differential field
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
equipped with a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
operator. Let ''K'' be a differential field with derivation operator ∂.
*A differential polynomial in ''x'' is a polynomial in the formal expressions ''x'', ∂''x'', ∂
2''x'', ... with coefficients in ''K''.
*The order of a non-zero differential polynomial in ''x'' is the largest ''n'' such that ∂
''n''''x'' occurs in it, or −1 if the differential polynomial is a constant.
*The separant ''S''
''f'' of a differential polynomial of order ''n''≥0 is the derivative of ''f'' with respect to ∂
''n''''x''.
*The field of constants of ''K'' is the subfield of elements ''a'' with ∂''a''=0.
*In a differential field ''K'' of nonzero
characteristic ''p'', all ''p''th powers are constants. It follows that neither ''K'' nor its field of constants is
perfect, unless ∂ is trivial. A field ''K'' with derivation ∂ is called differentially perfect if it is either of characteristic 0, or of characteristic ''p'' and every constant is a ''p''th power of an element of ''K''.
*A differentially closed field is a differentially perfect differential field ''K'' such that if ''f'' and ''g'' are differential polynomials such that ''S''
''f''≠ 0 and ''g''≠0 and ''f'' has order greater than that of ''g'', then there is some ''x'' in ''K'' with ''f''(''x'')=0 and ''g''(''x'')≠0. (Some authors add the condition that ''K'' has characteristic 0, in which case ''S''
''f'' is automatically non-zero, and ''K'' is automatically perfect.)
*DCF
''p'' is the theory of differentially closed fields of characteristic ''p'' (where ''p'' is 0 or a prime).
Taking ''g''=1 and ''f'' any ordinary
separable polynomial In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial.
This concept is closely rel ...
shows that any differentially closed field is
separably closed In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
. In characteristic 0 this implies that it is algebraically closed, but in characteristic ''p''>0 differentially closed fields are never algebraically closed.
Unlike the complex numbers in the theory of algebraically closed fields, there is no natural example of a differentially closed field.
Any differentially perfect field ''K'' has a differential closure, a
prime model
In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into an ...
extension, which is differentially closed. Shelah showed that the differential closure is unique up to isomorphism over ''K''. Shelah also showed that the prime differentially closed field of characteristic 0 (the differential closure of the rationals) is not
minimal; this was a rather surprising result, as it is not what one would expect by analogy with algebraically closed fields.
The theory of DCF
''p'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and
model complete In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding.
Equivalently, every first-order formula is equivalent to a universal formula.
This notion was introduced by Abraham Robinson ...
(for ''p''=0 this was shown by Robinson, and for ''p''>0 by ).
The theory DCF
''p'' is the
model companion In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding.
Equivalently, every first-order formula is equivalent to a universal formula.
This notion was introduced by Abraham Robinson ...
of the theory of differential fields of characteristic ''p''. It is the model completion of the theory of differentially perfect fields of characteristic ''p'' if one adds to the language a symbol giving the ''p''th root of constants when ''p''>0. The theory of differential fields of characteristic ''p''>0 does not have a model completion, and in characteristic ''p''=0 is the same as the theory of differentially perfect fields so has DCF
0 as its model completion.
The number of differentially closed fields of some infinite cardinality κ is 2
κ; for κ uncountable this was proved by , and for κ countable by Hrushovski and Sokolovic.
The Kolchin topology
The ''Kolchin topology'' on ''K''
m is defined by taking sets of solutions of systems of differential equations over ''K'' in ''m'' variables as basic closed sets. Like the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, the Kolchin topology is
Noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
.
A d-constructible set is a finite union of closed and open sets in the Kolchin topology. Equivalently, a d-constructible set is the set of solutions to a quantifier-free, or
atomic, formula with parameters in ''K''.
Quantifier elimination
Like the theory of algebraically closed fields, the theory DCF
0 of differentially closed fields of characteristic 0
eliminates quantifiers. The geometric content of this statement is that the projection of a d-constructible set is d-constructible. It also eliminates imaginaries, is complete, and model complete.
In characteristic ''p''>0, the theory DCF
p eliminates quantifiers in the language of differential fields with a unary function ''r'' added that is the ''p''th root of all constants, and is 0 on elements that are not constant.
Differential Nullstellensatz
The differential Nullstellensatz is the analogue in differential algebra of Hilbert's
nullstellensatz.
*A differential ideal or ∂-ideal is an ideal closed under ∂.
*An ideal is called radical if it contains all roots of its elements.
Suppose that ''K'' is a differentially closed field of characteristic 0. . Then Seidenberg's differential nullstellensatz states there is a bijection between
*Radical differential ideals in the ring of differential polynomials in ''n'' variables, and
*∂-closed subsets of ''K''
''n''.
This correspondence maps a ∂-closed subset to the ideal of elements vanishing on it, and maps an ideal to its set of zeros.
Omega stability
In characteristic 0 showed that the theory of differentially closed fields is
ω-stable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose model ...
and has
Morley rank In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.
Definition
Fix a theory ''T'' with a model ''M''. The Morley rank ...
ω.
In non-zero characteristic showed that the theory of differentially closed fields is not ω-stable, and showed more precisely that it is
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
but not
superstable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose model ...
.
The structure of definable sets: Zilber's trichotomy
Decidability issues
The Manin kernel
Applications
See also
*
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.
Overview
Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...
References
*
*
*
*
*
*
*{{citation, mr=1678539
, last=Wood, first= Carol , authorlink = Carol S. Wood
, chapter=Differentially closed fields, title= Model theory and algebraic geometry, pages=129–141
, series=Lecture Notes in Mathematics, volume= 1696, publisher= Springer, place= Berlin, year= 1998
, doi=10.1007/BFb0094671, doi-access=free, isbn=978-3-540-64863-5
Differential algebra
Model theory