GEOMETRIC AXIOMS FOR THE THEORY DCF 0 , m + 1 Omar Le on S anchez - PowerPoint PPT Presentation

GEOMETRIC AXIOMS FOR THE THEORY DCF 0 , m + 1 Omar Le´ on S´ anchez University of Waterloo March 24, 2011 (http://arxiv.org/abs/1103.0730) Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation In the 50’s Robinson showed that the class of existentially closed ordinary differential fields (of characteristic zero) is elementary. Then, in the 70’s, Blum gave elegant algebraic axioms: ( ord δ f > ord δ g ) → ( ∃ x f ( x ) = 0 ∧ g ( x ) � = 0) . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation In the 50’s Robinson showed that the class of existentially closed ordinary differential fields (of characteristic zero) is elementary. Then, in the 70’s, Blum gave elegant algebraic axioms: ( ord δ f > ord δ g ) → ( ∃ x f ( x ) = 0 ∧ g ( x ) � = 0) . In 1998, Pierce and Pillay gave axioms of DCF 0 in terms of algebraic varieties and their prolongation: K | = ACF 0 and ( V , W irreducible ) ∧ ( W ⊆ τ V ) ∧ ( W projects dominantly ) → ∃ ¯ x (¯ x , δ ¯ x ) ∈ W Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation In the 50’s Robinson showed that the class of existentially closed ordinary differential fields (of characteristic zero) is elementary. Then, in the 70’s, Blum gave elegant algebraic axioms: ( ord δ f > ord δ g ) → ( ∃ x f ( x ) = 0 ∧ g ( x ) � = 0) . In 1998, Pierce and Pillay gave axioms of DCF 0 in terms of algebraic varieties and their prolongation: K | = ACF 0 and ( V , W irreducible ) ∧ ( W ⊆ τ V ) ∧ ( W projects dominantly ) → ∃ ¯ x (¯ x , δ ¯ x ) ∈ W Geometric axiomatizations have been given for other theories ACFA , DCFA 0 , DCF p and SCH p , e . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation For existentially closed partial differential fields, DCF 0 , m , McGrail (2000) gave an algebraic axiomatization generalizing Blum’s. Other algebraic axiomatizations have been formulated by Yaffe (2001), Tressl (2005). Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation For existentially closed partial differential fields, DCF 0 , m , McGrail (2000) gave an algebraic axiomatization generalizing Blum’s. Other algebraic axiomatizations have been formulated by Yaffe (2001), Tressl (2005). A simple counterexample supplied by Hrushovski shows that the commutativity of the derivations imposes too many restrictions, so that ACF 0 together with ( V , W irreducible ) ∧ ( W ⊆ τ V ) ∧ ( W projects dominantly ) → ∃ ¯ x (¯ x , δ 1 ¯ x , . . . , δ m ¯ x ) ∈ W do not axiomatize DCF 0 , m . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Motivation For existentially closed partial differential fields, DCF 0 , m , McGrail (2000) gave an algebraic axiomatization generalizing Blum’s. Other algebraic axiomatizations have been formulated by Yaffe (2001), Tressl (2005). A simple counterexample supplied by Hrushovski shows that the commutativity of the derivations imposes too many restrictions, so that ACF 0 together with ( V , W irreducible ) ∧ ( W ⊆ τ V ) ∧ ( W projects dominantly ) → ∃ ¯ x (¯ x , δ 1 ¯ x , . . . , δ m ¯ x ) ∈ W do not axiomatize DCF 0 , m . Nonetheless, in 2010, Pierce formulated geometric axioms in arbitrary characteristic. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Our Approach We take a different approach and formulate geometric axioms for DCF 0 , m +1 in terms of a relative notion of prolongation. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Our Approach We take a different approach and formulate geometric axioms for DCF 0 , m +1 in terms of a relative notion of prolongation. In other words, we characterize DCF 0 , m +1 in terms of the geometry of DCF 0 , m . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Our Approach We take a different approach and formulate geometric axioms for DCF 0 , m +1 in terms of a relative notion of prolongation. In other words, we characterize DCF 0 , m +1 in terms of the geometry of DCF 0 , m . Theorem ( K , ∆ , D ) | = DCF 0 , m +1 if and only if 1 ( K , ∆) | = DCF 0 , m 2 For each pair of irreducible ∆-closed sets V and W such that W ⊆ τ D / ∆ V and W projects ∆-dominantly onto V , there is a K -point ¯ a ∈ V such that (¯ a , D ¯ a ) ∈ W . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. ∆-closed set means the zero set of ∆-polynomials, that is V ( f 1 , . . . , f s ). Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. ∆-closed set means the zero set of ∆-polynomials, that is V ( f 1 , . . . , f s ). θ ¯ x = ( θ 1 ¯ x , θ 2 ¯ x , . . . ) the set of algebraic indeterminates m · · · δ r 1 δ r m 1 x i , ordered w.r.t. the canonical ranking. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. ∆-closed set means the zero set of ∆-polynomials, that is V ( f 1 , . . . , f s ). θ ¯ x = ( θ 1 ¯ x , θ 2 ¯ x , . . . ) the set of algebraic indeterminates m · · · δ r 1 δ r m 1 x i , ordered w.r.t. the canonical ranking. For f ∈ K { ¯ x } , the Jacobian x ) := ( ∂ f x ) , ∂ f x ) , . . . , ∂ f df (¯ x (¯ x (¯ x (¯ x ) , 0 , 0 , . . . ) . ∂θ 1 ¯ ∂θ 2 ¯ ∂θ h ¯ Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. ∆-closed set means the zero set of ∆-polynomials, that is V ( f 1 , . . . , f s ). θ ¯ x = ( θ 1 ¯ x , θ 2 ¯ x , . . . ) the set of algebraic indeterminates m · · · δ r 1 δ r m 1 x i , ordered w.r.t. the canonical ranking. For f ∈ K { ¯ x } , the Jacobian x ) := ( ∂ f x ) , ∂ f x ) , . . . , ∂ f df (¯ x (¯ x (¯ x (¯ x ) , 0 , 0 , . . . ) . ∂θ 1 ¯ ∂θ 2 ¯ ∂θ h ¯ D : K → K another derivation commuting with ∆. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Notation ( K , ∆) field of characteristic zero with commuting derivations ∆ = { δ 1 , . . . , δ m } , K { ¯ x } the ∆-ring of ∆-polynomials. ∆-closed set means the zero set of ∆-polynomials, that is V ( f 1 , . . . , f s ). θ ¯ x = ( θ 1 ¯ x , θ 2 ¯ x , . . . ) the set of algebraic indeterminates m · · · δ r 1 δ r m 1 x i , ordered w.r.t. the canonical ranking. For f ∈ K { ¯ x } , the Jacobian x ) := ( ∂ f x ) , ∂ f x ) , . . . , ∂ f df (¯ x (¯ x (¯ x (¯ x ) , 0 , 0 , . . . ) . ∂θ 1 ¯ ∂θ 2 ¯ ∂θ h ¯ D : K → K another derivation commuting with ∆. f D the ∆-polynomial obtained by applying D to the coefficients of f . Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Definition Let τ D / ∆ : K { ¯ x } → K { ¯ x , ¯ y } be y + f D (¯ τ D / ∆ f (¯ x , ¯ y ) = df (¯ x ) · θ ¯ x ) τ D / ∆ is a derivation that extends D and commutes with ∆. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Definition Let τ D / ∆ : K { ¯ x } → K { ¯ x , ¯ y } be y + f D (¯ τ D / ∆ f (¯ x , ¯ y ) = df (¯ x ) · θ ¯ x ) τ D / ∆ is a derivation that extends D and commutes with ∆. Definition of D / ∆-prolongation Let V ⊆ K n be a ∆-closed set, then τ D / ∆ V ⊆ K 2 n is the ∆-closed set τ D / ∆ V = V ( f , τ D / ∆ f : f ∈ I ( V / K )) (1) I ( V / K ) := { f ∈ K { ¯ x } : f ( V ) = 0 } . Does τ D / ∆ V vary uniformly with V ? If I ( V / K ) is differentially generated by f 1 , . . . , f s then one only needs to check equation (1) for the f i ’s. Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Characterization of DCF 0 , m +1 Theorem 1 (L.S.) ( K , ∆ , D ) | = DCF 0 , m +1 if and only if 1 ( K , ∆) | = DCF 0 , m 2 For each pair of irreducible ∆-closed sets V and W such that W ⊆ τ D / ∆ V and W projects ∆-dominantly onto V , there is a K -point ¯ a ∈ V such that (¯ a , D ¯ a ) ∈ W . This uses a result of Kolchin about extending ∆-derivations. Expressing condition (2) in a first-order way is an issue: Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1

Characterization of DCF 0 , m +1 Theorem 1 (L.S.) ( K , ∆ , D ) | = DCF 0 , m +1 if and only if 1 ( K , ∆) | = DCF 0 , m 2 For each pair of irreducible ∆-closed sets V and W such that W ⊆ τ D / ∆ V and W projects ∆-dominantly onto V , there is a K -point ¯ a ∈ V such that (¯ a , D ¯ a ) ∈ W . This uses a result of Kolchin about extending ∆-derivations. Expressing condition (2) in a first-order way is an issue: Irreducibility of ∆-closed sets? Omar Le´ on S´ anchez Geometric Axioms for DCF 0 , m +1