Model Theory of Differential Fields

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54 DAVID MARKERdifferential fields, DF, is given by the axioms for fields of characteristic zero andthe axioms∀x ∀y δ(x + y) = δ(x) + δ(y),∀x ∀y δ(xy) = xδ(y) + yδ(x),which assert that δ is a derivation.If K is a differential field, we define K{X 1 , . . . , X n }, the ring of differentialpolynomials over K, to be the following polynomial ring in infinitely manyvariables:K [ X 1 , . . . , X n , δ(X 1 ), . . . , δ(X n ), . . . , δ m (X 1 ), . . . , δ m (X n ), . . . ] .We extend δ to a derivation on K{X 1 , . . . , X n } by setting δ(δ n (X i )) = δ n+1 (X i ).We say that K is an existentially closed differential field if, whenever f 1 , . . . ,f m ∈ K{X 1 , . . . , X n } and there is a differential field L extending K containinga solution to the system of differential equations f 1 = · · · = f m = 0, there isalready a solution in K. Robinson gave an axiomatization of the existentiallyclosed differential fields. Blum gave a simple axiomatization that refers only todifferential polynomials in one variable.If f ∈ K{X 1 , . . . , X n } \ K, the order of f is the largest m such that δ m (X i )occurs in f for some i. If f is a constant, we say f has order −1.Definition. A differential field K is differentially closed if, whenever f, g ∈K{X}, g is nonzero and the order of f is greater than the order of g, there isa ∈ K such that f(a) = 0 and g(a) ≠ 0.In particular, any differentially closed field is algebraically closed.For each m and d 0 and d 1 we can write down an L-sentence φ m,d0,d 1thatasserts that if f is a differential polynomial of order m and degree at most d 0and g is a nonzero differential polynomial of order less than m and degree atmost d 1 , then there is a solution to f(X) = 0 and g(X) ≠ 0. For example, φ 2,1,1is the formula∀a 0 ∀a 1 ∀a 2 ∀a 3 ∀b 0 ∀b 1 ∀b 2(a3 ≠ 0 ∧ (b 0 ≠ 0 ∨ b 1 ≠ 0 ∨ b 2 ≠ 0)−→ ∃x (a 3 δ(δ(x)) + a 2 δ(x) + a 1 x + a 0 = 0 ∧ b 2 δ(x) + b 1 x + b 0 ≠ 0) ) .The L-theory DCF is axiomatized by DF and the set of axioms φ m,d0,d 1, forall m, d 0 and d 1 . The models of DCF are exactly the differentially closed fields.It is not hard to show that if f, g ∈ K{X} are as above, then there is L ⊇ Kcontaining a solution to the system f(X) = 0 and Y g(X) − 1 = 0. Indeed wecould take L to be the fraction field of K{X}/P , where P is a minimal differentialprime ideal with f ∈ P . Iterating this construction shows that any differentialfield can be extended to a differentially closed field. Thus any existentially closedfield is differentially closed.
MODEL THEORY OF DIFFERENTIAL FIELDS 55The next theorem of Blum shows that the converse holds (see [Marker et al.1996] for the proof).Theorem 1.1. The theory DCF has quantifier elimination and hence is modelcomplete.Corollary 1.2. (i) DCF is a complete theory.(ii) A differential field is existentially closed if and only if it is differentiallyclosed.Proof. (i) The rational numbers with the trivial derivation form a differentialsubfield of any differentially closed field. If K 0 and K 1 are models of DCF and φis a quantifier free sentence, then there is a quantifier free sentence ψ such thatDCF |= φ ←→ ψ. But K i |= ψ if and only if Q |= ψ. Hence K 0 |= φ if and onlyif K 1 |= φ and DCF is complete.(ii) We already remarked that every existentially closed field is differentiallyclosed. Suppose K is differentially closed. Suppose f 1 = · · · = f m = 0 is a systemof polynomial differential equations solvable in an extension L of K. We can findK 1 an extension of L which is differentially closed. By model completeness K isan elementary submodel of K 1 . Since there is a solution in K 1 there is a solutionin K.□Pierce and Pillay [1998] have given a more geometric axiomatization of DCF.Suppose K is a differential field and V ⊆ K n is an irreducible algebraic varietydefined over K. Let I(V ) ⊂ K[X 1 , . . . , X n ] be the ideal of polynomials vanishingon V and let f 1 , . . . , f m generate I(V ). If f = ∑ a i1,...,i mX i11 · · · Xim m , let f δ =∑δ(ai1,...,i m)X i11 · · · Xim m . The tangent bundle T (V ) can be identified with thevarietyT (V ) ={(x, y) ∈ K 2n : x ∈ V ∧n∑j=1}∂f iy j (x) = 0 for i = 1, . . . , m∂X jWe define the first prolongation of V to be the algebraic variety{}n∑V (1) = (x, y) ∈ K 2n ∂f i: x ∈ V ∧ y j (x) + fi δ (x) = 0 for i = 1, . . . , m .∂X jj=1If V is defined over the constant field C, then each fi δ vanishes, and V (1) is T (V ).In general, for a ∈ V , the vector space T a (V ) = {b : (a, b) ∈ T (V )} acts regularlyon V a(1) = {b : (a, b) ∈ V (1) }, making V (1) a torsor under T (V ). It is easy to seethat (x, δ(x)) ∈ V (1) for all x ∈ V . Thus the derivation is a section of the firstprolongation.Theorem 1.3. For K be a differential field, the following statements are equivalent:(i) K is differentially closed.
Page 4 and 5: 56 DAVID MARKER(ii) K is existentia
Page 6 and 7: 58 DAVID MARKERThe existence of pri
Page 8 and 9: 60 DAVID MARKERTheorem 3.7. (i) (Pi
Page 10 and 11: 62 DAVID MARKERvariety of A is isom

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