DOCSLIB.ORG

Hardy Fields, O-Minimal Structures and Connections to Hilbert 16

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 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.
Load more

Recommended publications
  • N° 92 Séminaire De Structures Algébriques Ordonnées 2016
    N° 92 Séminaire de Structures Algébriques Ordonnées 2016 -- 2017 F. Delon, M. A. Dickmann, D. Gondard, T. Servi Février 2018 Equipe de Logique Mathématique Prépublications Institut de Mathématiques de Jussieu – Paris Rive Gauche Universités Denis Diderot (Paris 7) et Pierre et Marie Curie (Paris 6) Bâtiment Sophie Germain, 5 rue Thomas Mann – 75205 Paris Cedex 13 Tel. : (33 1) 57 27 91 71 – Fax : (33 1) 57 27 91 78 Les volumes des contributions au Séminaire de Structures Algébriques Ordonnées rendent compte des activités principales du séminaire de l'année indiquée sur chaque volume. Les contributions sont présentées par les auteurs, et publiées avec l'agrément des éditeurs, sans qu'il soit mis en place une procédure de comité de lecture. Ce séminaire est publié dans la série de prépublications de l'Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu–Paris Rive Gauche (CNRS -- Universités Paris 6 et 7). Il s'agit donc d'une édition informelle, et les auteurs ont toute liberté de soumettre leurs articles à la revue de leur choix. Cette publication a pour but de diffuser rapidement des résultats ou leur synthèse, et ainsi de faciliter la communication entre chercheurs. The proceedings of the Séminaire de Structures Algébriques Ordonnées constitute a written report of the main activities of the seminar during the year of publication. Papers are presented by each author, and published with the agreement of the editors, but are not refereed. This seminar is published in the preprint series of the Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu– Paris Rive Gauche (CNRS -- Universités Paris 6 et 7).
    [Show full text]
  • Computing the Homology of Semialgebraic Sets. II: General Formulas Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto
    Computing the Homology of Semialgebraic Sets. II: General formulas Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto To cite this version: Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto. Computing the Homology of Semialgebraic Sets. II: General formulas. Foundations of Computational Mathematics, Springer Verlag, 2021, 10.1007/s10208-020-09483-8. hal-02878370 HAL Id: hal-02878370 https://hal.inria.fr/hal-02878370 Submitted on 23 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Computing the Homology of Semialgebraic Sets. II: General formulas∗ Peter B¨urgissery Felipe Cuckerz Technische Universit¨atBerlin Dept. of Mathematics Institut f¨urMathematik City University of Hong Kong GERMANY HONG KONG [email protected] [email protected] Josu´eTonelli-Cuetox Inria Paris & IMJ-PRG OURAGAN team Sorbonne Universit´e Paris, FRANCE [email protected] Abstract We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data.
    [Show full text]
  • On the Total Curvature of Semialgebraic Graphs
    ON THE TOTAL CURVATURE OF SEMIALGEBRAIC GRAPHS LIVIU I. NICOLAESCU 3 RABSTRACT. We define the total curvature of a semialgebraic graph ¡ ½ R as an integral K(¡) = ¡ d¹, where ¹ is a certain Borel measure completely determined by the local extrinsic geometry of ¡. We prove that it satisfies the Chern-Lashof inequality K(¡) ¸ b(¡), where b(¡) = b0(¡) + b1(¡), and we completely characterize those graphs for which we have equality. We also prove the following unknottedness result: if ¡ ½ R3 is homeomorphic to the suspension of an n-point set, and satisfies the inequality K(¡) < 2 + b(¡), then ¡ is unknotted. Moreover, we describe a simple planar graph G such that for any " > 0 there exists a knotted semialgebraic embedding ¡ of G in R3 satisfying K(¡) < " + b(¡). CONTENTS Introduction 1 1. One dimensional stratified Morse theory 3 2. Total curvature 8 3. Tightness 13 4. Knottedness 16 Appendix A. Basics of real semi-algebraic geometry 21 References 25 INTRODUCTION The total curvature of a simple closed C2-curve ¡ in R3 is the quantity Z 1 K(¡) = jk(s) jdsj; ¼ ¡ where k(s) denotes the curvature function of ¡ and jdsj denotes the arc-length along C. In 1929 W. Fenchel [9] proved that for any such curve ¡ we have the inequality K(¡) ¸ 2; (F) with equality if and only if C is a planar convex curve. Two decades later, I. Fary´ [8] and J. Milnor [17] gave probabilistic interpretations of the total curvature. Milnor’s interpretation goes as follows. 3 3 Any unit vector u 2 R defines a linear function hu : R ! R, x 7! (u; x), where (¡; ¡) denotes 3 the inner product in R .
    [Show full text]
  • THE SEMIALGEBRAIC CASE Today's Goal Tarski-Seidenberg Theorem
    17 THE SEMIALGEBRAIC CASE Today’s Goal Tarski-Seidenberg Theorem. For every n =1, 2, 3,...,every Lalg-definable subset of Rn can be defined by a quantifier-free Lalg-formula. n Thus every Lalg-definable subset of R is a finite boolean combination (i.e., finitely many intersections, unions, and comple- ments) of sets of the form n {(x1,...,xn) ∈ R | p(x1,...,xn) > 0} where p(x1,...,xn)isapolynomial with coefficients in R. These are called the semialgebraic sets. A function f: A ⊂ Rn → Rm is semialgebraic if its graph is a semialgebraic subset of Rn × Rm. 18 As for the semilinear sets, every semialgebraic set can be written as a finite union of the intersection of finitely many sets defined by conditions of the form p(x1,...,xn)=0 q(x1,...,xn) > 0 where p(x1,...,xn) and q(x1,...,xn) are polynomials with coefficients in R. Proof Strategy • Prove a geometric structure theorem that shows that any semialgebraic set can be decomposed into finitely many semialgebraic generalized cylinders and graphs. • Deduce quantifier elimination from this. 19 Thom’s Lemma. Let p1(X),...,pk(X) be polynomials in the variable X with R coefficients in such that if pj(X) =0then pj(X) is included among p1,...,pk. Let S ⊂ R have the form S = ∩jpj(X) ∗j 0 where ∗j is one of <, >,or=, then S is either empty, a point, or an open interval. Moreover, the (topological) closure of S is obtained by changing relaxing the sign conditions (changing < to ≤ and > to ≥. Note There are 3k such possible sets, and these form a partition of R.
    [Show full text]
  • Describing Convex Semialgebraic Sets by Linear Matrix Inequalities
    Describing convex semialgebraic sets by linear matrix inequalities Markus Schweighofer Université de Rennes 1 International Symposium on Symbolic and Algebraic Computation Korea Institute for Advanced Study, Seoul July 28-31, 2009 Introduction n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by linear matrix inequalities n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition.
    [Show full text]
  • Asymptotic Differential Algebra
    ASYMPTOTIC DIFFERENTIAL ALGEBRA MATTHIAS ASCHENBRENNER AND LOU VAN DEN DRIES Abstract. We believe there is room for a subject named as in the title of this paper. Motivating examples are Hardy fields and fields of transseries. Assuming no previous knowledge of these notions, we introduce both, state some of their basic properties, and explain connections to o-minimal structures. We describe a common algebraic framework for these examples: the category of H-fields. This unified setting leads to a better understanding of Hardy fields and transseries from an algebraic and model-theoretic perspective. Contents Introduction 1 1. Hardy Fields 4 2. The Field of Logarithmic-Exponential Series 14 3. H-Fields and Asymptotic Couples 21 4. Algebraic Differential Equations over H-Fields 28 References 35 Introduction In taking asymptotic expansions `ala Poincar´e we deliberately neglect transfinitely 1 − log x small terms. For example, with f(x) := 1−x−1 + x , we have 1 1 f(x) ∼ 1 + + + ··· (x → +∞), x x2 so we lose any information about the transfinitely small term x− log x in passing to the asymptotic expansion of f in powers of x−1. Hardy fields and transseries both provide a kind of remedy by taking into account orders of growth different from . , x−2, x−1, 1, x, x2,... Hardy fields were preceded by du Bois-Reymond’s Infinit¨arcalc¨ul [9]. Hardy [30] made sense of [9], and focused on logarithmic-exponential functions (LE-functions for short). These are the real-valued functions in one variable defined on neigh- borhoods of +∞ that are obtained from constants and the identity function by algebraic operations, exponentiation and taking logarithms.
    [Show full text]
  • Constructions for the Theorem of Bröcker–Scheiderer
    Universit´e de Rennes I Institut de Recherche Campus de Beaulieu Math´ematique de Rennes 35042 Rennes Cedex Unit´e associ´ee au C.N.R.S. France n◦ 305 Rapport de stage Constructions for the Theorem of Br¨ocker{Scheiderer effectu´e par Andreas Bernig Stage de DEA 1996=97 Soutenu le 24 juin 1997 Sous la responsabilit´e de: a` Rennes Louis Mah´e Remerciements Ce rapport de stage a ´et´e ´ecrit au cours de l'ann´ee universitaire 1996/1997. Je voudrais tout d'abord remercier a` M. Mah´e d'avoir accept´e d'^etre le responsable de ce stage. A tout moment, il a ´et´e pr^et a` m'aider en r´epondant a` mes questions, en m'expliquant les math´ematiques et en me donnant des conseils tr`es utiles. Outre M. Mah´e, plusieures personnes m'ont aid´e pendant ce stage. Je tiens surtout a` remercier M. Becker (Dortmund) qui m'a propos´e ce sujet. M. Coste m'a ´egalement r´epondu a` beaucoup de mes questions. J'ai profit´e des discussions int´eressantes avec M. Monnier sur nos travaux ou` nous avons trouv´e quelques points communs. Pour les corrections soigneuses je veux dire merci a` W. Hildebrand dont les connaissances d'anglais m'ont beaucoup servi. Last but not least, Anne m'a donn´e la joie de vivre, m'a sans cesse motiv´e et encourag´e tout au long de ce travail. Contents Remerciements 2 1 Introduction 3 2 Real Algebra 5 2.1 Ordered fields and real closed fields .
    [Show full text]
  • Real Algebraic Geometry Lecture Notes (11: 24/11/09)
    REAL ALGEBRAIC GEOMETRY LECTURE NOTES (11: 24/11/09) SALMA KUHLMANN Contents 1. Normal form of semialgebraic sets 1 2. Geometric version of Tarski-Seidenberg 3 3. Formulas in the language of real closed fields 4 1. Normal form of semialgebraic sets Let R be a fixed real closed field and n > 1. We consider 3 operations on subsets of Rn: (1) finite unions, (2) finite intersections, (3) complements. Definition 1.1. (i) The class of semialgebraic sets in Rn is defined to be the smallest class of subsets of Rn closed under operations (1), (2), (3), and which contains all sets of the form fx 2 Rn : f(x) ¡ 0g; where f 2 R[x] = R[x1; : : : ; xn] and ¡ 2 f>; >; =; 6=g. (ii) Equivalently a subset S ⊆ Rn is semialgebraic if and only if it is a finite boolean combination of sets of the form fx 2 Rn : f(x) > 0g; where f(x) 2 R[x]. (iii) Consider 8f (x) ¡ 0 > 1 1 < . (∗) S(x) := . :> fk(x) ¡k 0 with fi(x) 2 R[x]; ¡i 2 f>; >; =; 6=g. The set of solutions of S(x) is precisely the semialgebraic set 1 2 SALMA KUHLMANN k \ n S := fx 2 R : fi(x) ¡i 0g: i=1 The solution set S of a system (∗) is called a basic semialgebraic subset of Rn. (iv) Let f1; : : : ; fk 2 R[x] = R[x1; : : : ; xn]. A set of the form n Z(f1; : : : ; fk) := fx 2 R : f1(x) = ··· = fk(x) = 0g is called an algebraic set. (v) A subset of Rn of the form U(f) : = fx 2 Rn : f(x) > 0g; n U(f1; : : : ; fk) : = fx 2 R : f1(x) > 0; : : : ; fk(x) > 0g = U(f1) \···\U(fk) is called a basic open semialgebraic set.
    [Show full text]
  • Differential Algebra, Ordered Fields and Model Theory
    UMONS - Universit´ede Mons Facult´edes Sciences Differential algebra, ordered fields and model theory Quentin Brouette Une dissertation pr´esent´eeen vue de l'obtention du grade acad´emiquede docteur en sciences. Directrice de th`ese Composition du jury Franc¸oise Point Thomas Brihaye Universit´ede Mons Universit´ede Mons, secr´etaire Codirecteur de th`ese Piotr Kowalski Christian Michaux Universit´ede Wroclaw Universit´ede Mons Cedric´ Riviere` Universit´ede Mons Marcus Tressl Universit´ede Manchester Maja Volkov Universit´ede Mons, pr´esidente Septembre 2015 Contents Note de l'auteur iii Preface v 1 Model theory, differential fields and ordered fields 1 1.1 Model Theory . .1 1.2 Differential fields . .7 1.2.1 Differential rings and fields . .8 1.2.2 Differentially closed fields . 11 1.3 Ordered fields . 14 1.4 Ordered differential fields and CODF . 17 1.4.1 Definitions and generalities . 17 1.4.2 Definable Closure . 19 1.4.3 Definable Skolem functions and imaginaries . 20 2 Stellens¨atze 21 2.1 The nullstellensatz for ordered fields . 23 2.2 The nullstellensatz for ordered differential fields . 24 2.3 Prime real differential ideals . 26 2.4 The positivstellensatz . 28 2.4.1 Topological version . 28 2.4.2 Algebraic version . 29 2.5 Schm¨udgen'stheorem . 31 2.5.1 Schm¨udgen's theorem for the real field . 32 2.5.2 Endowing R with a structure of CODF . 32 2.5.3 Schm¨udgen's theorem for differential fields . 34 3 Differential Galois Theory 37 3.1 Picard-Vessiot Extensions . 39 i ii 3.1.1 Generalities on linear differential equations and Picard- Vessiot extensions .
    [Show full text]
  • An Introduction to Semialgebraic Geometry
    AN INTRODUCTION TO SEMIALGEBRAIC GEOMETRY Michel COSTE Institut de Recherche Math´ematiquede Rennes [email protected] Octobre 2002 Contents Introduction 3 1 Counting the real roots 5 1.1 Sturm’s theorem . ....................... 5 1.1.1 P without multiple root . ............... 5 1.1.2 P with multiple roots . ................... 6 1.1.3 A bound for the roots ................... 7 1.2 Real roots satisfying inequalities . ............... 7 1.2.1 One inequality ....................... 7 1.2.2 Several inequalities . ................... 9 1.2.3 Deciding the existence of a solution of a system of poly- nomial equations and inequalities . .......... 10 1.3 Systems with parameters . ................... 10 1.3.1 Tarski-Seidenberg . ................... 11 1.3.2 Systems with one equation . ............... 12 1.3.3 Example: polynomial of degree 4 . .......... 13 1.3.4 General systems ....................... 16 1.4 Another method for counting real roots .............. 17 1.4.1 Hermite’s method . ................... 17 1.4.2 Determination of the signature .............. 19 1.4.3 Principal subresultant coefficients . .......... 21 2 Semialgebraic sets 25 2.1 Stability properties . ....................... 25 2.1.1 Definition and first examples . ............... 25 2.1.2 Consequences of Tarski-Seidenberg principle . ...... 26 2.2 Semialgebraic functions ....................... 29 2.2.1 Definition and first properties ............... 29 2.2.2 TheLojasiewicz 8 inequality . ............... 30 1 2 CONTENTS 2.3 Decomposition of a semialgebraic set . .............. 32 2.3.1 Cylindrical algebraic decomposition . ......... 32 2.3.2 Construction of an adapted c.a.d. ............. 34 2.3.3 The c.a.d. algorithm . .................. 38 2.3.4 Connected components of semialgebraic sets . ..... 40 3 Triangulation of semialgebraic sets 41 3.1 Thom’s lemma .
    [Show full text]
  • TAUT SUBMANIFOLDS ARE ALGEBRAIC 1. Introduction An
    TAUT SUBMANIFOLDS ARE ALGEBRAIC QUO-SHIN CHI Abstract. We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s. 1. Introduction An embedding f of a compact, connected manifold M into Euclidean space Rn is taut if every nondegenerate (Morse) Euclidean distance function, 2 n Lp : M ! R;Lp(z) = d(f(z); p) ; p 2 R ; has β(M; Z2) critical points on M, where β(M; Z2) is the sum of the Z2-Betti numbers of M. That is, Lp is a perfect Morse function on M. A slight variation of Kuiper's observation in [7] gives that tautness can be rephrased by the property that (1.1) Hj(M \ B; Z2) ! Hj(M; Z2) is injective for all closed disks B ⊂ Rn and all 0 ≤ j ≤ dim(M). As a result, tautness is a conformal invariant, so that via stereographic projection we can reformulate the notion of tautness in the sphere Sn using the spherical distance functions. Another immediate consequence is that if B1 ⊂ B2, then (1.2) Hj(M \ B1) ! Hj(M \ B2) is injective for all j. Kuiper in [8] raised the question whether all taut submanifolds in Rn are real algebraic. We established in [4] that a taut submanifold in Rn is real algebraic in the sense that, it is a connected component of a real irreducible algebraic variety in the same ambient space, provided the submanifold is of dimension no greater than 4.
    [Show full text]
  • Computing the Homology of Basic Semialgebraic Sets in Weak
    Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time PETER BÜRGISSER, Technische Universität Berlin, Germany FELIPE CUCKER, City University of Hong Kong PIERRE LAIREZ, Inria, France We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data). CCS Concepts: • Theory of computation → Computational geometry; • Computing methodologies → Hybrid symbolic-numeric methods. Additional Key Words and Phrases: Semialgebraic geometry, homology, algorithm ACM Reference Format: Peter Bürgisser, Felipe Cucker, and Pierre Lairez. 2018. Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time. J. ACM 66, 1, Article 5 (October 2018), 30 pages. https://doi.org/10.1145/3275242 1 INTRODUCTION Semialgebraic sets (that is, subsets of Euclidean spaces defined by polynomial equations and in- equalities with real coefficients) come in a wide variety of shapes and this raises the problem of describing a given specimen, from the most primitive features, such as emptiness, dimension, or number of connected components, to finer ones, such as roadmaps, Euler-Poincaré characteristic, Betti numbers, or torsion coefficients. The Cylindrical Algebraic Decomposition (CAD) introduced by Collins [23] and Wüthrich [63] n 5 in the 1970’s provided algorithms to compute these features that worked within time sD 2O( ) where s is the number of defining equations, D a bound on their degree and n the dimension( ) of the ambient space.
    [Show full text]