On local interdefinability of (real and complex) analytic functions - PowerPoint PPT Presentation

On local interdefinability of (real and complex) analytic functions Tamara Servi Université Paris Diderot 1st April 2018

Topic of this talk: given two (real or complex) analytic functions f and g , understand whether f and g are locally interdefinable, and why. Example: Theorem [Bianconi ’97]. Let R exp = � R ; 0 , 1 , + , · , exp,< � and [ a , b ] ⊆ R . Then the function � sin x x ∈ [ a , b ] f ( x ) = 0 x / ∈ [ a , b ] is not definable (with parameters) in R exp , i.e. no arc of sine can be obtained from the following geometric construction: start with zerosets and positivity sets of exponential polynomials and close under finite unions and intersections, taking complements and projections . A converse: exp (as any continuous function and any open U ⊆ R n ) is definable in R sin = � R ; 0 , 1 , + , · , sin , < � , since Z = { x : sin ( 2 π x ) = 0 } . So definability in R sin does not correspond to geometric constructions. A better converse: no restriction of exp to a compact interval is definable in the “restricted” structure R sin ↾ = � R ; 0 , 1 , + , · , sin ↾ [ 0 , 1 ] , < � (which is a reduct of R an , hence o-minimal). Summing up: exp and sin are not locally interdefinable .

Remark. If we identify C and R 2 , then exp and sin ↾ [ 0 , 1 ] define complex exponentiation on the strip D = { z : Im ( z ) ∈ [ 0 , 1 ] } . Bianconi’s result: complex exponentiation is not definable, even locally, from real exponentiation. As for complex exp, for many holomorphic functions (for example, periodic functions), if we separate the real and imaginary part, then the notion of (real) definability that gives rise to geometric constructions is necessarily local (within the realm of o-minimal geometry). Some motivation. • R helps C : the model theory of interesting holomorphic functions is less well understood than that of (the restrictions of) their real and imaginary parts — Peterzil & Starchenko: complex analysis in an o-minimal setting. • Express local definability of holomorphic functions in terms of complex operations (first separate real and imaginary part, then patch them back together) — work started by Wilkie. • C helps R : the geometry of sets definable via real analytic functions is better understood if we can access definably their holomorphic extensions (Weierstrass Preparation and quantifier elimination).

In this talk: [JKS14] Jones, Kirby, S., Local interdefinability of Weierstrass elliptic functions , JIMJ, 2014. [JKLS18] Jones, Kirby, Le Gal, S., On local definability of holomorphic functions , submitted, 2018. [LSV18] Le Gal, S., Vieillard-Baron, Isomorphic quasianalytic classes and definability , in preparation, 2018.

Local definability Definition (after A. Wilkie) . Let K = R or C . “Definable” means definable with parameters. • Let U ⊆ K n be open, g : U → K be (real/complex) analytic, ∆ ⊆ U be an open relatively compact box with rational corners. We call g ↾ ∆ a proper restriction of g . • Let F be a collection of (real/complex) analytic functions defined on open subsets of K n (for various n ∈ N ) and F ↾ be the collection of all proper restrictions of all functions in F . We let R F ↾ = � R ; 0 , 1 , + , · , <, F ↾ � be the expansion of the real field by the graphs of the functions in F ↾ (where we identify C with R 2 , if K = C ). (so R F ↾ is a reduct of R an ) • g : U → K is locally definable from F if all the proper restrictions of g are definable in R F ↾ . • F and G are not locally interdefinable if no f ∈ F is locally definable from G and no g ∈ G is locally definable from F . Remark. Let f be a (real/complex) analytic function. Then the Schwarz reflection f SR ( z ) := f ( z ) and the partial derivatives ∂ f ∂ z i ( z ) are locally definable from f .

Weierstrass elliptic functions In the spirit of Bianconi, consider the exponential map of the complex projective elliptic curve [ X : Y : Z ] ∈ P 2 ( C ) : Y 2 Z = 4 X 3 − aXZ 2 − bZ 3 � � E ( C ) = (for suitable a , b ∈ C ). More precisely, • Lattice: Λ = { n ω 1 + m ω 2 : n , m ∈ Z , ω 1 , ω 2 ∈ C lin. indep. / R } ⊆ C • Weierstrass ℘ -function wrto Λ : ℘ ( z ) = 1 � ( z − ω ) 2 − 1 1 � � z 2 + , ω 2 ω ∈ Λ \{ 0 } holomorphic on C \ Λ , periodic wrto Λ and differentially algebraic: ℘ ′ � 2 = 4 ( ℘ ) 3 − a ℘ − b , � ω ∈ Λ \{ 0 } ω − 4 , b = 140 � ω ∈ Λ \{ 0 } ω − 6 . with a = 60 � ℘ ( z ) : ℘ ′ ( z ) : 1 ∈ E ( C ) ⊆ P 2 ( C ) � � exp E : C ∋ z �− → is a homomorphism of complex Lie groups, with ker ( exp E ) = Λ . Question. What of local interdefinability of complex exp and a ℘ -function, or of two different ℘ -functions ℘ 1 and ℘ 2 ?

Orthogonality of Weierstrass ℘ -functions Theorem 1 [JKS14]. • No ℘ -function is locally interdefinable with complex exp • Two ℘ -functions ℘ 1 and ℘ 2 are locally interdefinable iff ℘ 2 is isogenous to (i.e. ∃ α ∈ C × s.t. Λ 1 ⊆ α Λ 2 or Λ 1 ⊆ α Λ 2 ) either ℘ 1 or ℘ SR 1 • More generally, let F 1 , F 2 be two disjoint sets of ℘ -functions. Then ℘ ∈ F 2 is locally definable from F 1 ∪ { exp } iff ∃ ˜ ℘ ∈ F 1 s.t. ℘ is isogenous to either ˜ ℘ or ℘ SR (we say that ℘ and ˜ ˜ ℘ are ISR-equivalent ) • Furthermore, suppose that the ℘ -functions in F 1 ∪ F 2 are pairwise non-ISR-equivalent and that X ⊆ R n . Let R 1 = R ( F 1 ∪{ exp } ) ↾ and R 2 = R F 2 ↾ . Then X is definable in both R 1 and R 2 iff X is semialgebraic. Keypoint (Ax’s theorem): Let ϕ = { ϕ j } m j = 1 be a Q -linearly independent set of power series without constant term. Then tr.deg C ( { ϕ j ( z ) , exp ( ϕ j ( z )) : j = 1 , . . . , m } ) ≥ m + 1 [Brownawell & Kubota, 1977]. An Ax-type functional transcendence statement for exp and finitely many pairwise non-isogenous ℘ -functions, applied to linearly independent sets of power series without constant term. Theorem 1 says: not only are these functions algebraically independent, but they are also pairwise orthogonal wrto local definability.

An application: proving that certain functions are transcendental Remark. let F be the set of all ℘ -functions and let f : ( a , b ) − → R be a transcendental real analytic function locally definable in R F . Then the function g ( x ) = exp ( f ( log x )) is transcendental: otherwise f = log ◦ g ◦ exp is definable also in R exp , and hence f is algebraic, by Theorem 1. A counting application. Let f be as above and, for q = a b ∈ Q , let H ( q ) = max ( | a | , | b | ) . Then there exist constants c , γ > 0 (depending only on f ) such that # { ( log p , log q ) ∈ Γ ( f ) : p , q ∈ Q , H ( p ) , H ( q ) ≤ T } ≤ c ( log T ) γ . Proof. • Enough to count the pairs ( p , q ) ∈ Q 2 ∩ Γ ( g ) . • Show that g is definable in a model-complete reduct R of R Pfaff . • Apply a counting theorem for transcendental curves definable in R , due to Jones & Thomas.

Proof of Theorem 1: ingredients Remark 1. • ℘ 2 = ℘ SR = ⇒ ℘ 2 locally definable from ℘ 1 (actually, Λ 2 = Λ 1 ) 1 • α ∈ C × and Λ 1 = α Λ 2 = ⇒ ℘ 2 ( z ) = α 2 ℘ 1 ( α z ) • Λ 1 ⊆ Λ 2 = ⇒ ℘ 2 is an elliptic function periodic wrto Λ 1 = ⇒ ℘ 2 is a rational combination of ℘ 1 and ℘ ′ 1 (known fact) Hence, if ℘ 2 is ISR-equivalent to ℘ 1 (i.e. ∃ α ∈ C × s.t. Λ 1 ⊆ α Λ 2 or Λ 1 ⊆ α Λ 2 ), then ℘ 2 is locally definable from ℘ 1 . Remark 2. Let F be a collection of holomorphic functions and g / ∈ F be a holomorphic function. If g is obtained from functions in F by composition or by extracting implicit functions, then clearly g is locally definable from F . Ingredient 1 [Wilkie ’08]. Let z 0 be suitably generic . Then g is locally definable from F in a neighbourhood of z 0 iff g is obtained from functions in F and polynomials by finitely many applications of Schwarz reflection , differentiation , composition and extracting implicit functions . Ingredient 2 [Brownawell & Kubota ’77]. For i = 1 , . . . , n , let: f i = exp or f i = ℘ i , (with ℘ i , ℘ j non-isogenous), K i = CM-field of f i ( Q or a quadratic extension of Q ), ϕ i = { ϕ i , j } m i j = 1 a K i -lin. indep. set of power series ∈ z C � z � . Then n � tr.deg C ( { ϕ i , j ( z ) , f i ( ϕ i , j ( z )) : i = 1 , . . . , n , j = 1 , . . . , m i } ) ≥ m i + 1 i = 1

Proof of Theorem 1: an easy case Let ℘ 1 , ℘ 2 be non-isogenous ℘ -functions such that Λ 1 = Λ 1 and Λ 2 = Λ 2 . Suppose for a contradiction that ℘ 2 is locally definable from ℘ 1 . Wilkie’s theorem (i.e. ℘ 2 is obtained from ℘ 1 by differentiation, composition, implicit function ) + properties of ℘ -functions (e.g. differential algebraicity, the group structure on the elliptic curve ), imply that ℘ 2 is generically implicitly definable from ℘ 1 : around a suitably chosen z 0 , for some m ∈ N , there is an ( m + 1 ) -tuple g = ( g 1 ( z ) , . . . , g m + 1 ( z )) of holomorphic functions, with g 1 ( z ) = z and g 2 ( z ) = ℘ 2 ( z ) , such that the ( 2 m + 2 ) -tuple { g i ( z ) , ℘ 1 ( g i ( z )) } m + 1 i = 1 satisfies a nonsingular system of m polynomial equations . In particular, { g i ( z ) , ℘ 1 ( g i ( z )) } m + 1 � � tr.deg C ≤ ( 2 m + 2 ) − m = m + 2 . i = 1 By Ax’s theorem [BK77], applied to ℘ 1 , ℘ 2 , with ϕ 1 = g ( z ) , ϕ 2 = { z } , { g i ( z ) , ℘ 1 ( g i ( z )) } m + 1 � � tr.deg C ≥ | ϕ 1 | + | ϕ 2 | + 1 = ( m + 1 )+ 1 + 1 = m + 3 � i = 1