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 +1 of real valued functions (with the usual ring operations). A germ g 2 G is continuous, respectively differentiable, if it is the germ of a continuous, respectively differentiable, function ]a; +1[! R for some a 2 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 2 H are differentiable, and h0 2 H for all h 2 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 +1.A Laurent series at +1 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 2 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 +1. Let g be a germ at +1 and let k k−1 −1 −1 G = ak x + ak−1x + ::: + a0 + a−1x + ::: 2 R((x )): Then g has the asymptotic expansion G at +1 if for every N 2 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 2 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 2 R f (x) = 0; g1(x) > 0;:::; gk (x) > 0 ; where f ; g1;:::; gk 2 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 2 R the fiber n Ax := y 2 R (x; y) 2 A is finite. Then there is N 2 N such that Ax has at most N m elements for every x 2 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 n2 such that each Mn n N consists of subsets of R such that the following properties hold: (a) A; B 2 Mn =) A [ B; A \ B; A n B 2 Mn. (b) A 2 Mm; B 2 Mn =) A × B 2 Mm+n. n+1 n (c) A 2 Mn+1 =) π(A) 2 Mn where π : R ! R is the projection on the first n coordinates. n (d) A ⊂ R semialgebraic =) A 2 Mn. It is called o-minimal if additionally: (e) A 2 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 2 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 2 R the fiber n Ax := y 2 R (x; y) 2 A is finite. Then there is N 2 N such that Ax has at most N m elements for every x 2 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 8 n < p(x); x 2 [−1; 1] ; f (x) = if : 0; x 2= [−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 +1 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) 2 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.