 Differentially closed field

In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues for differential equations of algebraically closed fields for polynomial equations.
Contents
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, ∂^{2}x, ...
 The order of a nonzero 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 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 perfect.
 A differentially closed field is a differentially perfect 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 the field 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 nonzero, and K is automatically perfect.)
 DCF_{p} is the theory of differentially closed fields of characteristic p (0 or a prime).
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 DCF_{p} is complete and model complete (for p=0 this was shown by Robinson, and for p>0 by Wood (1973)). The theory DCF_{p} 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 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 Shelah (1973), 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, the Kolchin topology is Noetherian.
A dconstructible set is a finite union of closed and open sets in the Kolchin topology. Equivalently, a dconstructible set is the set of solutions to a quantifierfree, 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 dconstructible set is dconstructible. 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 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 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 Blum () showed that the theory of differentially closed fields is ωstable and has Morley rank ω. In nonzero characteristic Wood (1973) showed that the theory of differentially closed fields is not ωstable, and Shelah (1973) showed more precisely that it is stable but not superstable.
The structure of definable sets: Zilber's trichotomy
Decidability issues
The Manin kernel
Applications
See also
References
 Marker, David (2000), "Model theory of differential fields", Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., 39, Cambridge: Cambridge Univ. Press, pp. 53–63,, MR1773702, http://www.msri.org/publications/books/Book39/files/dcf.pdf
 Robinson, Abraham (1959), "On the concept of a differentially closed field.", Bull. Res. Council Israel Sect. F 8F: 113–128, MR0125016
 Sacks, Gerald E. (1972), "The differential closure of a differential field", Bull. Amer. Math. Soc. 78 (5): 629–634, doi:10.1090/S000299041972129690, MR0299466, http://www.ams.org/bull/19727805/S000299041972129690/
 Shelah, Saharon (1973), "Differentially closed fields", Israel J. Math. 16 (3): 314–328, doi:10.1007/BF02756711, MR0344116
 Wood, Carol (1973), "The Model Theory of Differential Fields of Characteristic p ≠ 0", Proceedings of the American Mathematical Society (Proceedings of the American Mathematical Society, Vol. 40, No. 2) 40 (2): 577–584, doi:10.2307/2039417, JSTOR 2039417
 Wood, Carol (1976), "The model theory of differential fields revisited", Israel Journal of Mathematics 25 (3–4): 331, doi:10.1007/BF02757008
 Wood, Carol (1998), "Differentially closed fields", Model theory and algebraic geometry, Lecture Notes in Math., 1696, Berlin: Springer, pp. 129–141, doi:10.1007/BFb0094671, ISBN 9783540648635, MR1678539
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