Differentially closed field
Encyclopedia
In mathematics
, a differential field K is differentially closed if every finite system of differential equation
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.
Taking g=1 and f any ordinary separable polynomial
shows that any differentially closed field is separably closed. In characteristic 0 this implies that it is algebraically closed, but in characteristic p>0 differentially closed fields are never algebraically closed (or perfect), as the differential must vanish on any pth power.
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
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 DCFp is complete
and model complete (for p=0 this was shown by Robinson, and for p>0 by ).
The theory DCFp is the model companion 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 pth 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 DCF0 as its model completion.
The number of differentially closed fields of some infinite cardinality κ is 2κ; for κ uncountable this was proved by , and for κ countable by Hrushovski and Sokolovic.
, the Kolchin topology is Noetherian
.
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.
. 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 DCFp eliminates quantifiers in the language of differential fields with a unary function r added that is the pth root of all constants, and is 0 on elements that are not constant.
Suppose that K is a differentially closed field of characteristic 0. . Then Seidenberg's differential nullstellensatz states there is a bijection between
This correspondence maps a ∂-closed subset to the ideal of elements vanishing on it, and maps an ideal to its set of zeros.
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
but not superstable.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a differential field K is differentially closed if every finite system of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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
- p is 0 or a prime number, and is the characteristic of a field.
- A differential polynomial in x is a polynomial in x, ∂x, ∂2x, ...
- The order of a non-zero differential polynomial in x is the largest n such that ∂nx occurs in it, or −1 if the differential polynomial is a constant.
- The separant Sf of a differential polynomial of order n≥0 is the derivative of f with respect to ∂nx.
- The field of constants of a differential field is the subfield of elements a with ∂a=0.
- A field with derivation is called differentially perfect if the field of constants is perfectPerfect fieldIn algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
. - A differentially closed field is a differentially perfect field K such that if f and g are differential polynomials such that Sf≠ 0 and g≠0 and f has order greater than that of g, then there is some x in the field with f(x)=0 and g(x)≠0. (Some authors add the condition that K has characteristic 0, in which case Sf is automatically non-zero, and K is automatically perfect.)
- DCFp is the theory of differentially closed fields of characteristic p (0 or a prime).
Taking g=1 and f any ordinary separable polynomial
Separable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
shows that any differentially closed field is separably closed. In characteristic 0 this implies that it is algebraically closed, but in characteristic p>0 differentially closed fields are never algebraically closed (or perfect), as the differential must vanish on any pth power.
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
Prime model
In mathematics, and in particular model theory, a prime model is a model which 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 .- Cardinality :In contrast with the notion of saturated model,...
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
Minimal model
*In theoretical physics, the term minimal model usually refers to a special class of conformal field theories that generalize the Ising model, or to some closely related representations of the Virasoro algebra...
; this was a rather surprising result, as it is not what one would expect by analogy with algebraically closed fields.
The theory of DCFp is complete
Complete theory
In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent...
and model complete (for p=0 this was shown by Robinson, and for p>0 by ).
The theory DCFp is the model companion 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 pth 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 DCF0 as its model completion.
The number of differentially closed fields of some infinite cardinality κ is 2κ; 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 topologyZariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
, the Kolchin topology is Noetherian
Noetherian topological space
In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets...
.
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
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
, formula with parameters in K.
Quantifier elimination
Like the theory of algebraically closed fields, the theory DCF0 of differentially closed fields of characteristic 0 eliminates quantifiersQuantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification...
. 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 DCFp eliminates quantifiers in the language of differential fields with a unary function r added that is the pth 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 Kn.
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 and has Morley rankMorley 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...
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
Stable theory
In 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 models are too complicated to classify, and to classify all models in the cases...
but not superstable.