1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Hardy Fields, O-minimal Structures and connections to Hilbert 16
Tobias Kaiser
Universit¨at Passau
16. September 2018
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
1. Hardy Fields
Hardy fields are named after Godfrey Harold Hardy. The context is asymptotics of real valued functions.
Setting: Let G be the ring of germs at +∞ of real valued functions (with the usual ring operations). A germ g ∈ G is continuous, respectively differentiable, if it is the germ of a continuous, respectively differentiable, function ]a, +∞[→ R for some a ∈ R.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Definition: A Hardy field is a subring H of G such that H is a field, all h ∈ H are differentiable, and h0 ∈ H for all h ∈ H.
Examples: (1) The field R. −1 (2) The field R(x) = R(x ) of rational functions. −1 (3) The field Rhhx ii of convergent Laurent series at +∞.A Laurent series at +∞ is a series of the form
k k−1 −1 −2 ak x + ak−1x + ... + a1x + a0 + a−1x + a−2x + ...
with real coefficients aj .
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Remark: Let H be a Hardy field and let h ∈ H. Then either eventually h(t) > 0 or eventually h(t) = 0 or eventually h(t) < 0. Setting h > 0 if the first case holds, the Hardy field is made an ordered field.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Asymptotics and quasianalyticity −1 By R((x )) we denote the field of formal Laurent series at +∞. Let g be a germ at +∞ and let
k k−1 −1 −1 G = ak x + ak−1x + ... + a0 + a−1x + ... ∈ R((x )).
Then g has the asymptotic expansion G at +∞ if for every N ∈ Z k N N g(x) − ak x + ... + aN x ) = o(x ). We write than g ∼ G. Note that G is uniquely determined if it exists.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Let R be a subring of G and assume that every g ∈ R has an asymptotic expansion T (g). Assume that R is quasianalytic; i.e. the map −1 T : R → R((x )), g → T (g), is injective. Then elements of R fulfil the above dichotomy. In particular R is a domain. If additionally R consists of differentiable germs and is closed under differentiation with T (g 0) = T (g)0 then the quotient field of R is a Hardy field.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
2. O-Minimal Structures
O-minimal structures enrich algebraic geometry by important concepts of analysis. Real algebraic geometry Real algebraic geometry considers semialgebraic sets. Definition: n A semialgebraic set in R is a finite union of sets of the form
n x ∈ R f (x) = 0, g1(x) > 0,..., gk (x) > 0 ,
where f , g1,..., gk ∈ R[X ] = R[X1,..., Xn].
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Examples: (1) Real varieties. (2) Complex varieties. (3) Intervals, balls.
Tameness:
I A semialgebraic set has only finitely many connected components, each of them semialgebraic.
I Semialgebraic triangulation.
I Semialgebraic stratification.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
I Uniform finiteness: m+n m Let A ⊂ R be semialgebraic such that for every x ∈ R the fiber n Ax := y ∈ R (x, y) ∈ A
is finite. Then there is N ∈ N such that Ax has at most N m elements for every x ∈ R .
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
O-minimal structures Definition: A structure M is a sequence Mn n∈ such that each Mn n N consists of subsets of R such that the following properties hold: (a) A, B ∈ Mn =⇒ A ∪ B, A ∩ B, A \ B ∈ Mn.
(b) A ∈ Mm, B ∈ Mn =⇒ A × B ∈ Mm+n. n+1 n (c) A ∈ Mn+1 =⇒ π(A) ∈ Mn where π : R → R is the projection on the first n coordinates. n (d) A ⊂ R semialgebraic =⇒ A ∈ Mn. It is called o-minimal if additionally:
(e) A ∈ M1 ⇐⇒ A is a finite union of intervals and points.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
n A subset A of R is definable in M if A ∈ Mn. A function n m f : A → R where A ⊂ R is definable in M if its graph is definable in M. Tameness: Axiomatically, the subsets of R definable in an o-minimal structure have finitely many connected components. But one obtains the same tameness properties as above:
I A definable set has only finitely many connected components, each of them definable.
I Definable triangulation.
I Definable stratification.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
I Uniform finiteness: m+n m Let A ⊂ R be definable such that for every x ∈ R the fiber n Ax := y ∈ R (x, y) ∈ A
is finite. Then there is N ∈ N such that Ax has at most N m elements for every x ∈ R .
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Examples of o-minimal structures: I The pure field R; the definable sets are exactly the semialgebraic ones.
I Ran, the structure generated by the restricted analytic n functions. A function f : R → R is restricted analytic if there is a power series p that converges on a neighbourhood of [−1, 1]n such that
n p(x), x ∈ [−1, 1] , f (x) = if 0, x ∈/ [−1, 1]n.
Equivalently: The structure generated by the bounded subanalytic sets.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
I Rexp, the structure generated by the global exponential function.
I Ran,exp, the structure generated by the restricted analytic functions and the global exponential function. The latter structure is of most importance regarding applications (for example in diophantine geometry) since all elementary functions (with the necessary restrictions) are definable:
I Polynomials
I Power functions
I Exponential function
I Logarithmic function
I Restricted sine and cosine
I Arctangent
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Fact: The germs at +∞ of unary functions definable in an o-minimal structure form a Hardy field.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
3. Hilbert 16
One of the most important questions in the theory of analytic vector fields is Hilbert’s 16th-problem, Part 2. Traditionally it is split up into three questions of increasing difficulty.
I Does a polynomial vector field in the plane have only finitely many limit cycles?
I Is the number of limit cycles uniformely bounded by the degree of the polynomials?
I How does such an upper bound look like?
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Polynomial vector field in the plane:
x0 = p(x, y) y 0 = q(x, y)
where p(x, y), q(x, y) ∈ R[X , Y ]. Trajectory: Solution of the differential equation
Cycle: Periodic trajectory
Limit cycle: Isolated cycle
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
So far (after a long history of false proofs) only the first question (also known as Dulac’s problem) has been answered. The positive answer has been established independently by Ilyashenko and Ecalle´ around 1990.
Approach of Ilyashenko: Poincar´e-mapsand transition maps at singularities of polycycles
Polycycle: Finite set of singularities connected by trajectories
Poincar´e-map: First-return-map. Fixed points correspond to cycles, isolated fixed points to limit cycles. Poincar´e-mapsare finite compositions of transition maps and analytic functions.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
The first (but already highly non-trivial) case is given by transition maps at hyperbolic singularities:
The linear part of the vector field has (after a suitable coordinate transformation) the form
λ 0 0 −µ
where λ ≥ µ > 0. The transition map has then an asymptotic expansion X k+lα Pk,l (log z)z , k,l∈N
where Pk,l ∈ C[T ] und α := λ/µ (Dulac 1923). If α is irrational (non-resonant case), then the polynomials Pk,l are constant; i.e. no logarithmic terms occur.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Ilyashenko has shown that the transition maps belong to a quasianalytic class. The functions of this class are holomorphic on so-called quadratic domains in the Riemann surface of the logarithm and allow the above asymptotic expansion on these sets. The quasianalyticity follows from a Phragm´en-Lindel¨of principle. From this the proof of Dulac’s problem in the case of hyperbolic singularities is deduced.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
4. Why O-Minimality?
I The proofs of Ilyashenko and Ecalle´ of Dulac’s problem in the general case are very hard to access. If one could show that transition maps at arbitrary singularities (together with analytic functions) generate an o-minimal structure, the proof of Dulac’s problem would follow immediately.
I The hope is to use the uniform finiteness property of o-minimal structures to contribute to the case of families of polynomial vector fields.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
5. Results
Old Result (K., Rolin, Speissegger) The transition maps at non-resonant hyperbolic saddles generate an o-minimal structure.
The logarithmic perturbation terms in the resonant case make the functions much more difficult to handle!!!
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
New Result (Galal, K., Speissegger) There is a Hardy field that contains all transistion maps at hyperbolic singularities and all unary functions definable in the o-minimal structure Ran,exp. Corollary Non-oscillation of transition maps at hyperbolic singularities and all elementary functions (with the necessary restrictions).
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
Ingredients: (A) Study the holomorphic extandability of germs of unary functions definable in Ran,exp. Wilkie has characterized those germs of unary functions definable in Ran,exp which have a definable holomorphic extension to a sector. K. and Speissegger have recently described the maximal extension of the germ of a unary function f definable in Ran,exp that is infinitely increasing to a biholomorphic (definable) function F : V → W . This maximal extension is described by two invariants of the germ of f , the exponential height and the level. The exponential height measures the size of the domain V , the level measures the size of the codomain W .
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
The exponential height measures the complexity of a germ/term. It is given by
] exp-terms - ] log-terms
needed to build the function/term. Examples √ (1) eh x · exp( log x) = 0. (2) eh x + exp(−x) = 1. (3) eh log x = −1.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
The level measures the growth of an infinitely increasing germ. There are k, s ∈ Z such that
logk ◦f ∼ logk−s .
Then s is the level of f . (Rosenlicht; Marker, Miller)
Examples level(x) = 0, level(exp) = 1, level(log) = −1.
Remark We have eh(f ) ≥ level(f ).
Example We have eh x + exp(−x) = 1, level x + exp(−x) = 0.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
(B) Transseries as asymptotic expansions. Power series (or variants thereof) are not sufficient to describe the asymptotic behaviour of functions in the Hardy field we aim for. We need transseries. A transseries is a formal series of the form
1/3 2 x xex +ex−x +x5/2−x(log x)2+log x+6+x−1+x−2+x−3+...+e−x +e−e .
Momomials are for example: x, ex , ex−x1/3 , log x.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16 1. Hardy Fields 2. O-minimal Structures 3. Hilbert 16 4. Why o-minimality? 5. Results 6. Outlook
6. Outlook
Ongoing Research:
I Define germs of algebras of functions in several variables.
I Show stability under various operations (such as blowing ups).
I Establish o-minimality by a normalization procedure.
Tobias Kaiser Universit¨at Passau Hardy Fields, O-minimal Structures and connections to Hilbert 16