Analytic difference fields Elimination of field quantifiers and - PowerPoint PPT Presentation

Analytic difference fields Elimination of field quantifiers and Ax-Kochen-Eršov principle Silvain Rideau École Normale Supérieure, Orsay Paris-Sud 11 1 / 22

Take Martin’s talk, insert analytic everywhere. 2 / 22

Thank you 3 / 22

More seriously Let K be a complete valued field. evaluated at c . hence f can be evaluated at c 4 / 22 ▸ Let ( a i ) i ∈ N be a sequence in K , ∑ i a i converges if and only if a i → 0. ▸ Let f = ∑ i a i X i ∈ O [[ X ]] and c ∈ M , then a i c i → 0 and hence f can be ▸ Let f = ∑ i a i X i ∈ O ⟨ X ⟩ = { ∑ i a i X i ∶ a i → 0 } and c ∈ O then a i c i → 0 and ▸ Similarly, any f ∈ O ⟨ X ⟩[[ Y ]] can be evaluated at any c ∈ O ∣ X ∣ × M ∣ Y ∣ .

Fields with analytic structure Let A be a Noetherian ring, I an ideal and suppose that A is complete and Example 5 / 22 separated in its I -adic topology. Let A m , n ∶ = A ⟨ X ⟩[[ Y ]] where ∣ X ∣ = m and ∣ Y ∣ = n , A ∶ = ⋃ m , n A m , n . Definition: Field with (separated) ( A , I ) -analytic structure A field with ( A , I ) -analytic structure is a valued field K with ring morphisms i m , n ∶ A m , n → O O m × M n such that: ▸ i 0 , 0 ( I ) ⊆ M ; ▸ i m , n ( X i ) ∶ O m × M n → O is the i -th projection function; ▸ i m , n ( Y j ) ∶ O m × M n → O is the ( m + j ) -th projection function (followed by the inclusion M ⊆ O ); ▸ The i m , n are compatible with the obvious injections A m , n → A m + k , n + l . Any complete field K has a natural ( O , M ) -analytic structure.

Valued fields language Definition: Angular component maps An angular component map on a valued field K is a group morphism Example component map. 6 / 22 ac ∶ K ⋆ → k ⋆ such that: ac ∣ O ⋆ = res ∣ O ⋆ ▸ On K (( X )) , ∑ i > n a i X i ↦ a i where a i ≠ 0 is an angular component map. ▸ On Q p , ∑ i > n a i p i ↦ a i where a i ≠ 0 is also an angular component map. ▸ Any ℵ 1 -saturated valued field can be endowed with an angular

Valued fields language Definition: Angular component maps An angular component map on a valued field K is a group morphism Remark This is not the right language to consider mixed characteristic (or equicharacteristic p ) in. Angular component add new definable sets. There exists another language known as the RV language that does not have this flaw. 6 / 22 ac ∶ K ⋆ → k ⋆ such that: ac ∣ O ⋆ = res ∣ O ⋆ We will be considering valued fields in the following language L : ▸ sorts K , Γ , k ; ▸ the ring language on K and k ; ▸ the language of ordered abelian groups on Γ ; ▸ v ∶ K → Γ and ac ∶ K → k .

Analytic language Fix a ring A and an ideal I as previously. Definition of their domain); 7 / 22 Let L A ∶ = L ∪ A ∪ { Q } where ▸ f ∈ A m , n is a function symbol K m + n → K ; ▸ Q is a function symbol K 2 → K ; Any field K with analytic ( A , I ) -structure can be naturally endowed with an L A -structure: ▸ the symbols f ∈ A m , n are interpreted as i m , n ( f ) (extended by 0 outside ▸ Q ( x , y ) is interpreted as x / y when y is not 0 and 0 otherwise;

Henselian valued fields Definition that: Example Let T ac 8 / 22 A valued field K is said to be Henselian if for all P ∈ O [ X ] and a ∈ O such v ( P ( a )) > 2v ( P ′ ( a )) , then there exists b ∈ O such that: P ( b ) = 0 and v ( b − a ) = v ( P ( a )) − v ( P ′ ( a )) . ▸ Any K (( X )) is Henselian. ▸ Q p is Henselian. A , H , 0 , 0 be the L A -theory of equicharacteristic zero Henselian fields with ( A , I ) -analytic structure and angular components.

Completions Definition: Balls Definition: Spherically complete valued fields A valued field K is said to be spherically complete for every decreasing b i Proposition Every spherically complete valued field is Henselian. 9 / 22 Let K be a valued field, c ∈ K and γ ∈ v ( K ) . ▸ The open ball of radius γ around c is { x ∈ K ∶ v ( x − c ) > γ } . ▸ The closed ball of radius γ around c is { x ∈ K ∶ v ( x − c ) ≥ γ } . sequence of balls ( b i ) i ∈ I in K , there exists c ∈ ⋂ i ∈ I

Analytic results I Theorem: Cluckers-Lipshitz-Robinson... T ac Definition: Resplendent elimination of field quantifiers Remark The key property behind this result is Weierstrass preparation that allows us to reduce questions about 1-types in T ac considerations (and use field quantifier elimination result in Henselian fields). 10 / 22 A , H , 0 , 0 eliminates field quantifiers resplendently. An L A -theory T is said to eliminate field quantifiers resplendently if whenever L ′ is an enrichment of L A on k and Γ and T ′ ⊇ T is an L ′ -theory, then T ′ eliminates field quantifiers. A , H , 0 , 0 to purely algebraic

Analytic results II as ordered abelian groups with constants Resplendent versions of these statements hold. Remark Corollary: Analytic NIP Ax-Kochen-Eršov 11 / 22 Corollary: Analytic Ax-Kochen-Eršov Let T be the set of all L A -terms (from the sort K ) without free variables. Let K and L ⊧ T ac A , H , 0 , 0 then: ⎧ k ( K ) ≡ k ( L ) as rings with constants added for the ac ( t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ where t ∈ T ; Γ ( K ) ≡ Γ ( L ) ⎪ ⎪ K ≡ L ⇐ ⇒ ⎪ ⎪ ⎪ added for the v ( t ) where t ∈ T . ⎩ Let K ⊧ T ac A , H , 0 , 0 then: Th ( K ) is NIP ⇐ ⇒ Th ( k ( K )) , as a ring, is NIP .

And now for something completely different... 12 / 22

Isometries Definition Definition: Residually linearly closed An valued field with an isometry K is said to be residually linearly closed if solution. Definition: Enough Constants A valued field with an isometry K is said to have enough constants if for 13 / 22 Let K be a valued field. An isometry on K is an automorphism σ of valued fields such that for all x ∈ K , v ( σ ( x )) = v ( x ) . ▸ Any valued field automorphism induces an automorphism of the residue field σ k ( x ) . ▸ We will write Fix ( K ) for the fixed field. k ( x ) = b has a for every non zero tuple a ∈ k and b ∈ k , the equation ∑ i a i σ i every x ∈ K there exists y ∈ Fix ( K ) such that v ( x ) = v ( y ) .

A new notion of Henselianity Proposition Every residually linearly closed spherically complete field with an 14 / 22 We will write σ ( a ) for the tuple a ,...,σ n ( a ) . Definition: σ -Henselianity A valued field with an isometry K is said to be σ -Henselian if for any P ∈ O [ X 0 ,..., X n ] and a ∈ O such that: v ( P ( σ ( a ))) > 2min i { v ( ∂ P ( σ ( a )))} , ∂ X i then there exists b ∈ O such that: P ( σ ( b )) = 0 and v ( b − a ) = v ( P ( σ ( a ))) − min i { v ( ∂ P ( σ ( a )))} . ∂ X i isometry is σ -Henselian.

Yet another language Definition Any valued field with an isometry can be naturally endowed with an Definition Let T ac trivially valued fields with an isometry and enough constants. Remark Models of T ac 15 / 22 Let L σ ∶ = L ∪ { σ,σ k } where σ is a function symbol K → K and σ k is a function symbol k → k . L σ -structure. σ − H , 0 , 0 be the L σ -theory of equicharacteristic zero σ -Henselian non σ − H , 0 , 0 are residually linearly closed.

Valued difference results Theorem: Azgin-van den Dries, 2010 Corollary: Valued difference NIP Ax-Kochen-Eršov as ordered abelian groups. as difference rings; Corollary: Valued difference Ax-Kochen-Eršov T ac 16 / 22 σ − H , 0 , 0 eliminates field quantifiers resplendently. Let K and L ⊧ T ac σ − H , 0 , 0 then: ⇒ { k ( K ) ≡ k ( L ) Γ ( K ) ≡ Γ ( L ) K ≡ L ⇐ Let K ⊧ T ac σ − H , 0 , 0 then: Th ( K ) is NIP ⇐ ⇒ Th ( k ( K )) , as a difference ring, is NIP .

And now... In stereo. 17 / 22

Analytic valued fields with an isometry f Definition also holds. Definition . An isometry of K (as an analytic field) 18 / 22 Let K be a field with ( A , I ) -analytic structure. For all m , n choose an automorphism A m , n A m , n → f σ ↦ is an isometry of K such that for all f ∈ A and x ∈ K , σ ( f ( x )) = f σ ( σ ( x )) If all the automorphisms are the identity then we are just asking that σ is an automorphism of K as a field with ( A , I ) -analytic structure. Let L A ,σ ∶ = L A ∪ L σ .

Some differentiation Definition: Linear approximation i Remark approximated by its derivatives. 19 / 22 Let K be a valued field with an isometry, f ∶ K n → K , d ∈ K n , a ∈ K and γ ∈ v ( K ) . We say that d is a linear approximation of f (at prolongations) around a with radius γ if for all b ∈ K such that v ( b − a ) > γ and ε ∈ K such that v ( ε ) > γ : v ( f ( σ ( b + ε )) − f ( σ ( b )) − ∑ d i σ i ( ε )) > min i { v ( d i )} + v ( ε ) ▸ Not very compatible with sum, product, composition... ▸ If f is continuous differentiable around a then it is linearly ▸ Very much akin to the Jacobian property of Cluckers-Lipshitz.