Model Theory of Differential Fields
Model Theory of Differential Fields Model Theory of Differential Fields
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
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54 DAVID MARKERdifferential fields, DF, is given by the axioms for fields <strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> differential equations f 1 = · · · = f m = 0, there isalready a solution in K. Robinson gave an axiomatization <strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> f is greater than the order <strong>of</strong> 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 <strong>of</strong> order m and degree at most d 0and g is a nonzero differential polynomial <strong>of</strong> 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 <strong>of</strong> axioms φ m,d0,d 1, forall m, d 0 and d 1 . The models <strong>of</strong> 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 <strong>of</strong> 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.
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