An Introduction to Semialgebraic Geometry
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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). -
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. -
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 . -
Algebraic, Geometric, and Topological Methods in Optimization
ALGEBRAIC,GEOMETRIC, AND TOPOLOGICAL METHODS IN OPTIMIZATION Jesús A. De Loera University of California, Davis 2019 — Joint Mathematical Meetings, Baltimore 1 This talk is about USING ALGEBRA, GEOMETRY and TOPOLOGY in in OPTIMIZATION research Due to time I will focus onLINEAR OPTIMIZATION... THE STORY OF THIS LECTURE -Optimization is a vibrant branch of Applied Mathematics. Its goal is to maximizing or minimizing some objective function relative to a set of possible solutions! E.g., Maximum profit, optimal arrangement, minimal error, etc. 2 - This talk is about USING ALGEBRA, GEOMETRY and TOPOLOGY in in OPTIMIZATION research Due to time I will focus onLINEAR OPTIMIZATION... THE STORY OF THIS LECTURE Optimization is a vibrant branch of Applied Mathematics. Its goal is to maximizing or minimizing some objective function relative to a set of possible solutions! E.g., Maximum profit, optimal arrangement, minimal error, etc. 2 - Due to time I will focus onLINEAR OPTIMIZATION... THE STORY OF THIS LECTURE Optimization is a vibrant branch of Applied Mathematics. Its goal is to maximizing or minimizing some objective function relative to a set of possible solutions! E.g., Maximum profit, optimal arrangement, minimal error, etc. This talk is about USING ALGEBRA, GEOMETRY and TOPOLOGY in in OPTIMIZATION research 2 - THE STORY OF THIS LECTURE Optimization is a vibrant branch of Applied Mathematics. Its goal is to maximizing or minimizing some objective function relative to a set of possible solutions! E.g., Maximum profit, optimal arrangement, minimal error, etc. This talk is about USING ALGEBRA, GEOMETRY and TOPOLOGY in in OPTIMIZATION research Due to time I will focus onLINEAR OPTIMIZATION.. -
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. -
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. -
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. -
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 . -
Real Geometric Invariant Theory and Ricci Soliton Metrics on Two-Step Nilmanifolds
Real Geometric Invariant Theory and Ricci Soliton Metrics on Two-step Nilmanifolds Michael R. Jablonski A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics. Chapel Hill 2008 Approved by Advisor: Patrick B. Eberlein Reader: James Damon Reader: Shrawan Kumar Reader: Michael Taylor Reader: Jonathan Wahl ABSTRACT MICHAEL R. JABLONSKI: Real Geometric Invariant Theory and Ricci Soliton Metrics on Two-step Nilmanifolds (Under the direction of Patrick B. Eberlein) In this work we study Real Geometric Invariant Theory and its applications to left-invariant geometry of nilpotent Lie groups. We develop some new results in the real category that distinguish GIT over the reals from GIT over the complexes. Moreover, we explore some of the basic relationships between real and complex GIT over projective space to obtain analogues of the well-known relationships that previously existed in the affine setting. This work is applied to the problem of finding left-invariant Ricci soliton metrics on two-step nilpotent Lie groups. Using our work on Real GIT, we show that most two-step nilpotent Lie groups admit left- invariant Ricci soliton metrics. Moreover, we build many new families of nilpotent Lie groups which cannot admit such metrics. ii To the Lord my God. iii ACKNOWLEDGEMENTS I would like to thank my thesis advisor, Dr. Patrick Eberlein, who introduced me to many interesting and worthwhile avenues of mathematics; his encouragement to not limit myself to one narrow field has made all the difference. -
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. -
Quantitative Properties of Real Algebraic Singularities
Ph. D thesis : Quantitative properties of real algebraic singularities Lionel F. Alberti Directeur de th`ese : Bernard Mourrain Codirecteur de th`ese : Georges Comte September 22, 2008 Contents 1 Introduction 4 1.1 Fran¸cais................................ 4 1.2 English ................................ 7 2 Fast and certified topology computations for planar curves 11 2.1 Notationsanddefinitions . 13 2.2 Overview ............................... 14 2.3 Regulardomains ........................... 15 2.4 Simplysingulardomains. 19 2.4.1 TopologicalDegree . 19 2.4.2 Counting the number of branches . 21 2.4.3 Conic structure and connection algorithm . 22 2.5 Isolatingtheinterestingpoints . 24 2.5.1 Subdivisionmethod . 25 2.5.2 Rational univariate representation . 27 2.6 Examples ............................... 28 3 Triangulating smooth algebraic varieties 31 3.1 Backgroundonmeshing implicit surfaces. 31 3.2 Algebraicingredients. 34 3.2.1 Representationofpolynomials . 34 3.2.2 Univariatesolver . 35 3.3 Towardaguaranteedmethod . 36 3.3.1 Descriptionofthealgorithm. 36 3.3.2 Complexityanalysis . 40 3.3.3 Singularities.......................... 42 3.4 Experimentation ........................... 43 4 Whitney stratifications, Transversality, and Triangulations 45 4.1 Stratification and topological stability . ... 51 4.1.1 Notations and background on stratifications . 51 4.1.2 An extension of the moving the wall theorem . 57 4.2 Measuringtransversality. 67 4.3 Conicstructure ............................ 72 1 4.3.1 The classical setup of the conic structure theorem . 72 4.3.2 Astableversionofconicstructure . 75 4.4 Relating Euclidean distance and transversality . .... 79 4.5 A tentative triangulation procedure . 84 5 A sweeping method to triangulate surfaces in three-space 93 5.1 Notations ............................... 94 5.1.1 Thecontourcurve .. .. .. .. .. .. .. .. .. 95 5.1.2 Characteristicpointsonthesurface . 96 5.2 Topologyofthex-criticalsections. -
Real Algebraic Geometry Reading Group Topics MPI Leipzig July 3–7, 2017
Real algebraic geometry reading group topics MPI Leipzig July 3{7, 2017. The following are proposed topics for talks to be delivered by the participants of the workshop. Further suggestions from you the participants are also welcome! Topology of real algebraic varieties • Constructing real algebraic varieties with interesting topology This talk should briefly describe the construction methods of Harnack, Hilbert, and Gudkov [Utk78] and an explanation of the classification of planar sextics. Then give a description of Viro's patchworking construction and theorem [Vir01] from the point of view of tropical geometry [IMS09]. This could also include Haas' patchworking theorem for curves [BIMS15, Chapter 3.3]. • Topology of real K3 surfaces This talk should describe Nikulin's classification of real K3 surfaces and planar sextics up to rigid isotopy via lattice theory [Nik80]. If time permits the speaker could also mention the classification of 1-nodal planar sextics by Itenberg by similar methods [Ite94]. • Topology of real algebraic space curves While the topology of planar curves has been extensively studied, the sudy of the topology of real algebraic curves in RP3 remained relatively unexamined. This talk should address the rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for small degree and genus [Bj11, MO16] and of canonically embedded curves of genus four [DZ99]. Convex algebraic geometry • Spectrahedral shadows Spectrahedral shadows are the feasible sets of semidefinite programming and thus the qustion of their characterisation is of high interest. A large class of semi-algebraic convex sets are in fact spectrahedral shadows as shown in [HN09, HN10].