DOCSLIB.ORG

When Is the Product of Intervals Also an Interval?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
When Is the Product of Intervals Also an Interval? When is the product of intervals also an interval? Olga Kosheleva1 and Piet G. Vroegindeweij2 1Department of Electrical and Computer Engineering University of Texas at El Paso El Paso, TX 79968, USA email [email protected] 2Koenestraat 10 NL 3958 XG Amerongen, the Netherlands Abstract Interval arithmetic is based on the fact that for intervals on the real line, the element-wise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is an element-wise product or a sum of two intervals always an interval? In this paper, we analyze this problem in a general algebraic setting: we need the corresponding algebraic structures to have (related) addition, multi- plication, and order; thus, we consider (consistently) ordered rings. We describe all consistently ordered rings for which the element-wise product and sum of two intervals are always intervals. 1 Introduction Intervals are important. Measurements are never 100% precise. As a result, if the measurement result is, say, 100, this does not mean that the actual value is exactly 100, it rather means that the actual value must be within an interval [100 ¡ ¢; 100 + ¢], where ¢ is an upper bound on the measurement error (this upper bound is usually supplied by the manufacturer of the measuring instrument). For example, if ¢ = 1, then the actual value can be equal to any number within the interval [99; 101]. Similarly, if we, say, measure the total charge to determine the total amount of electrons in a given beam, and the measurements are not accurate enough, we do not get the exact number of electrons, but only the interval of possible values, such as [99; 101], meaning that there can be either 99, or 100, or 101 electrons (i.e., that [99; 101] = f99; 100; 101g). 1 In both cases, after the measurement, we get not the exact value of the measured quantity, but only the interval of its possible values: a = [a; a] = fa 2 X j a · a · ag: (1) In the ¯rst example, the set X of possible actual values of the measured quantity is a set R of all real numbers; in the second example, X coincides with the set of all integers. Intervals on the real line: Traditional interval arithmetic. Often, we need to process the measured data. If the measured quantities are real numbers, then the measurement results can be expressed by intervals on the real line. To process such \real-interval" data, we can use interval arithmetic originally proposed by R. E. Moore (for recent surveys, see, e.g., [2, 3]). In general, in data processing, we are interested in the quantity b = f(a1; : : : ; an) that is related, in a known way, with directly measured quantities a1; : : : ; an. To describe possible values of a, we measure the quantities a1; : : : ; an, and use the resulting intervals a1 = [a1; a1];:::; an = [an; an] of possible values of ai to describe the set B of possible values of b as B = fb j b = f(a1; : : : ; an) for some a1 2 [a1; a1]; : : : ; an 2 [an; an]g: (2) The resulting set B is called the range of the function f(a1; : : : ; an) on the intervals a1;:::; an, or the result of applying the function f(a1; : : : ; an) to the intervals a1;:::; an. In particular, if the data processing operation f(a1; : : : ; an) consists of simply applying an arithmetic operation + or ¤, i.e., if f(a1; a2) = a1 + a2 or f(a1 ¤ a2) = a1 ¤ a2, then we get the element-wise de¯nitions of the sum and of the product of two intervals: [a1; a1]+[a2; a2] = fb j b = a1 +a2 for some a1 2 [a1; a1] and a2 2 [a2; a2]g; (3) [a1; a1] ¤ [a2; a2] = fb j b = a1 ¤ a2 for some a1 2 [a1; a1] and a2 2 [a2; a2]g: (4) For intervals on the real line, thus de¯ned element-wise sum and product are also intervals (and the formulas for computing the endpoints of these intervals form the basis of interval arithmetic). Intervals on the set of integers are more di±cult to handle than in- tervals on the real line. For intervals on the set of all integers, we can also de¯ne element-wise addition and multiplication by using formulas (3) and (4), but these integer-intervals are more computationally di±cult to handle because the product of two integer-intervals is not always an interval: e.g., the element- wise product of two integer-intervals [1; 2] = f1; 2g and [1; 3] = f1; 2; 3g is equal to [1; 2] ¤ [1; 3] = f1; 2; 3; 4; 6g; this product is not an integer-interval. 2 A natural question. A natural question is: How can we describe all algebraic structures for which the product of two intervals is always an interval? To formulate this problem in precise terms, we need to have an algebraic struc- ture with addition, multiplication, and order that are (in some reasonable way) consistent. In the next section, we show that natural consistency demands lead to a notion of a consistently ordered ring. Thus, in algebraic terms, we get the following problem: Given an arbitrary consistently ordered ring K, we can de¯ne another structure of interval objects of the form fa 2 K j a · a · ag, with addi- tion and multiplication de¯ned element-wise. We want to describe all the rings for which this new interval structure is closed under addition and multiplication. 2 Motivation for the Following De¯nitions We need two consistent operations and one relation. We want to for- mulate our problem in general algebraic terms. Since our question is about addition and multiplication of intervals, this structure must contain addition, multiplication, and the order (so that we will be able to de¯ne intervals); we must also require that these operations be consistent with each other. Addition. First, we need an operation called addition. We will assume that addition + is commutative and associative, that there exists an element 0 for which 0 + a = a for all a, and that for every element a, there is an inverse element ¡a for which a + (¡a) = 0. In other words, we will assume that our structure K is an Abelian (commutative) group under addition. Multiplication. Second, we need an operation called multiplication. It is natural to require that multiplication is associative and distributive with respect to addition (i.e., (a + b) ¤ c = a ¤ c + b ¤ c and a ¤ (b + c) = a ¤ c + b ¤ c). These properties de¯ne a ring. So, we will consider rings. Comment. Multiplication of real numbers has an additional property: it is com- mutative (a¤b = b¤a). However, we do not want to impose this restriction on the general structure, because there are meaningful computation-related examples of non-commutative ordered rings: e.g., a ring of all n £ n real-valued square matrices over A (with A ¸ 0 i® A is non-negative de¯nite) is non-commutative. Order. In order to de¯ne intervals on a ring, we must have an order. The above matrix example shows that the order need not be total (linear); therefore, in this paper, we will consider the general case of a (partial) order. This order 3 must be consistent with + and ¤. There are several reasonable requirements that express this consistency: ² First, there must exist a positive element a > 0. ² Second, the order must be shift-invariant: if a < b, then for every c, we must have a + c < b + c. ² Third, the sum and the product of two positive numbers must be positive. ² Fourth, if a ¤ b is positive and a is positive, then b must be positive too, and if both a ¤ b and b are positive, then a must be positive. There is a ¯fth natural property that does not need to be speci¯cally postulated for linear orders, but which is needed for partial orders: the product of two non- zero numbers must be non-zero. (If we have a linear order, then every non-zero number is either positive or negative, and this property simply follows from the third one.) These ¯ve properties are true for all three examples that we had considered so far: for the ring of all real numbers, for the ring of all integers (with natural order in both examples), and for the ring of all n £ n square matrices. The notions described by these properties are well known in algebra: an object that satis¯es the ¯rst three properties is called an ordered ring; an ordered ring that satis¯es the fourth property as well is called a consistently ordered ring, and a ring that satis¯es the additional ¯fth property is called a ring with no divisors of zero. In these terms, in the following text, we will consider consistently ordered rings with no divisors of 0. We are now ready to introduce formal de¯nitions: 3 De¯nitions and the Main Result 3.1 De¯nitions of the algebraic structure: consistently or- dered rings, intervals, operations with intervals De¯nition 1. (see, e.g., [4]) ² A ring K is a set of elements with two binary operations + and ¤ (called addition and multiplication) that satis¯es the following three properties: { K is an Abelian (commutative) group under addition; { multiplication is associative; { the right- and left-distributive laws hold, i.e., a ¤ (b + c) = a ¤ b + a ¤ c and (a + b) ¤ c = a ¤ c + b ¤ c.
Load more

Recommended publications
  • 14. Least Upper Bounds in Ordered Rings Recall the Definition of a Commutative Ring with 1 from Lecture 3
    14. Least upper bounds in ordered rings Recall the definition of a commutative ring with 1 from lecture 3 (Definition 3.2 ). Definition 14.1. A commutative ring R with unity consists of the following data: a set R, • two maps + : R R R (“addition”), : R R R (“multiplication”), • × −→ · × −→ two special elements 0 R = 0 , 1R = 1 R, 0 = 1 such that • ∈ 6 (1) the addition + is commutative and associative (2) the multiplication is commutative and associative (3) distributive law holds:· a (b + c) = ( a b) + ( a c) for all a, b, c R. (4) 0 is an additive identity. · · · ∈ (5) 1 is a multiplicative identity (6) for all a R there exists an additive inverse ( a) R such that a + ( a) = 0. ∈ − ∈ − Example 14.2. The integers Z, the rationals Q, the reals R, and integers modulo n Zn are all commutative rings with 1. The set M2(R) of 2 2 matrices with the usual matrix addition and multiplication is a noncommutative ring with 1 (where× “1” is the identity matrix). The set 2 Z of even integers with the usual addition and multiplication is a commutative ring without 1. Next we introduce the notion of an integral domain. In fact we have seen integral domains in lecture 3, where they were called “commutative rings without zero divisors” (see Lemma 3.4 ). Definition 14.3. A commutative ring R (with 1) is an integral domain if a b = 0 = a = 0 or b = 0 · ⇒ for all a, b R. (This is equivalent to: a b = a c and a = 0 = b = c.) ∈ · · 6 ⇒ Example 14.4.
    [Show full text]
  • A Note on Embedding a Partially Ordered Ring in a Division Algebra William H
    proceedings of the american mathematical society Volume 37, Number 1, January 1973 A NOTE ON EMBEDDING A PARTIALLY ORDERED RING IN A DIVISION ALGEBRA WILLIAM H. REYNOLDS Abstract. If H is a maximal cone of a ring A such that the subring generated by H is a commutative integral domain that satisfies a certain centrality condition in A, then there exist a maxi- mal cone H' in a division ring A' and an order preserving mono- morphism of A into A', where the subring of A' generated by H' is a subfield over which A' is algebraic. Hypotheses are strengthened so that the main theorems of the author's earlier paper hold for maximal cones. The terminology of the author's earlier paper [3] will be used. For a subsemiring H of a ring A, we write H—H for {x—y:x,y e H); this is the subring of A generated by H. We say that H is a u-hemiring of A if H is maximal in the class of all subsemirings of A that do not contain a given element u in the center of A. We call H left central if for every a e A and he H there exists h' e H with ah=h'a. Recall that H is a cone if Hn(—H)={0], and a maximal cone if H is not properly contained in another cone. First note that in [3, Theorem 1] the commutativity of the hemiring, established at the beginning of the proof, was only exploited near the end of the proof and was not used to simplify the earlier details.
    [Show full text]
  • Algorithmic Semi-Algebraic Geometry and Topology – Recent Progress and Open Problems
    Contemporary Mathematics Algorithmic Semi-algebraic Geometry and Topology – Recent Progress and Open Problems Saugata Basu Abstract. We give a survey of algorithms for computing topological invari- ants of semi-algebraic sets with special emphasis on the more recent devel- opments in designing algorithms for computing the Betti numbers of semi- algebraic sets. Aside from describing these results, we discuss briefly the back- ground as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems. Contents 1. Introduction 1 2. Semi-algebraic Geometry: Background 3 3. Recent Algorithmic Results 10 4. Algorithmic Preliminaries 12 5. Topological Preliminaries 22 6. Algorithms for Computing the First Few Betti Numbers 41 7. The Quadratic Case 53 8. Betti Numbers of Arrangements 66 9. Open Problems 69 Acknowledgment 71 References 71 1. Introduction This article has several goals. The primary goal is to provide the reader with a thorough survey of the current state of knowledge on efficient algorithms for computing topological invariants of semi-algebraic sets – and in particular their Key words and phrases. Semi-algebraic Sets, Betti Numbers, Arrangements, Algorithms, Complexity . The author was supported in part by NSF grant CCF-0634907. Part of this work was done while the author was visiting the Institute of Mathematics and its Applications, Minneapolis. 2000 Mathematics Subject Classification Primary 14P10, 14P25; Secondary 68W30 c 0000 (copyright holder) 1 2 SAUGATA BASU Betti numbers. At the same time we want to provide graduate students who intend to pursue research in the area of algorithmic semi-algebraic geometry, a primer on the main technical tools used in the recent developments in this area, so that they can start to use these themselves in their own work.
    [Show full text]
  • Embedding Two Ordered Rings in One Ordered Ring. Part I1
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 2, 341-364 (1966) Embedding Two Ordered Rings in One Ordered Ring. Part I1 J. R. ISBELL Department of Mathematics, Case Institute of Technology, Cleveland, Ohio Communicated by R. H. Bruck Received July 8, 1965 INTRODUCTION Two (totally) ordered rings will be called compatible if there exists an ordered ring in which both of them can be embedded. This paper concerns characterizations of the ordered rings compatible with a given ordered division ring K. Compatibility with K is equivalent to compatibility with the center of K; the proof of this depends heavily on Neumann’s theorem [S] that every ordered division ring is compatible with the real numbers. Moreover, only the subfield K, of real algebraic numbers in the center of K matters. For each Ks , the compatible ordered rings are characterized by a set of elementary conditions which can be expressed as polynomial identities in terms of ring and lattice operations. A finite set of identities, or even iden- tities in a finite number of variables, do not suffice, even in the commutative case. Explicit rules will be given for writing out identities which are necessary and sufficient for a commutative ordered ring E to be compatible with a given K (i.e., with K,). Probably the methods of this paper suffice for deriving corresponding rules for noncommutative E, but only the case K,, = Q is done here. If E is compatible with K,, , then E is embeddable in an ordered algebra over Ks and (obviously) E is compatible with the rational field Q.
    [Show full text]
  • Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 4, October 1996 MODEL COMPLETENESS RESULTS FOR EXPANSIONS OF THE ORDERED FIELD OF REAL NUMBERS BY RESTRICTED PFAFFIAN FUNCTIONS AND THE EXPONENTIAL FUNCTION A. J. WILKIE 1. Introduction Recall that a subset of Rn is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form α~ Rn : p(α~)=0, { ∈ } α~ Rn:q(α~)>0 where p(~x), q(~x)aren-variable polynomials with real co- { ∈ } efficients. A map from Rn to Rm is called semi-algebraic if its graph, considered as a subset of Rn+m, is so. The geometry of such sets and maps (“semi-algebraic geometry”) is now a widely studied and flourishing subject that owes much to the foundational work in the 1930s of the logician Alfred Tarski. He proved ([11]) that the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic. (A familiar simple instance: the image of a, b, c, x R4 :a=0andax2 +bx+c =0 {h i∈ 6 } under the projection map R3 R R3 is a, b, c R3 :a=0andb2 4ac 0 .) Tarski’s result implies that the× class→ of semi-algebraic{h i∈ sets6 is closed− under≥ first-} order logical definability (where, as well as boolean operations, the quantifiers “ x R ...”and“x R...” are allowed) and for this reason it is known to ∃ ∈ ∀ ∈ logicians as “quantifier elimination for the ordered ring structure on R”. Immedi- ate consequences are the facts that the closure, interior and boundary of a semi- algebraic set are semi-algebraic.
    [Show full text]
  • With Squares Positive and an /-Superunit Can Be Embedded in a Unital /-Prime Lattice-Ordered Ring with Squares Positive
    PROCEEDINGSof the AMERICANMATHEMATICAL SOCIETY Volume 121, Number 4, August 1994 THE UNITABILITY OF /-PRIME LATTICE-ORDEREDRINGS WITH SQUARESPOSITIVE MA JINGJING (Communicated by Lance W. Small) Abstract. It is shown that an /-prime lattice-ordered ring with squares positive and an /-superunit can be embedded in a unital /-prime lattice-ordered ring with squares positive. In [1, p. 325] Steinberg asked if an /-prime /-ring with squares positive and an /-superunit can be embedded in a unital /-prime /-ring with squares positive. In this paper we show that the answer is yes. If R is a lattice-ordered ring (/-ring) and a e R+, then a is called an /- element of R if b A c = 0 implies ab/\c = baAc = 0. Let T = T(R) = {a e R : \a\ is an /-element of R}. Then T is a convex /-subring of R, and R is a subdirect product of totally ordered T-T bimodules [2, Lemma 1]. R is an /-ring precisely when T = R. Throughout this paper T will denote the subring of /-elements of R. An element e > 0 of an /-ring R is called a superunit if ex > x and xe > x for each x e R+ ; e is an /-superunit if it is a superunit and an /-element. R is infinitesimal if x2 < x for each x in R+ . The /-ring R is called /-prime if the product of two nonzero /-ideals is nonzero and /-semiprime if it has no nonzero nilpotent /-ideals. An /-ring R is called squares positive if a2 > 0 for each a in R.
    [Show full text]
  • Convex Extensions of Partially Ordered Rings
    Convex Extensions of Partially Ordered Rings Niels Schwartz*) A series of lectures given at the conference "Géométrie algébrique et analytique réelle" Kenitra, Morocco, September 13 – 20, 2004 This series of lectures is about algebraic aspects of real algebraic geometry. Partially ordered rings (porings) occur very naturally in real algebraic geometry, as well as in other branches of mathematics. There is a rich classical theory of porings (the beginnings dating back to the 1930s and 1940s). To a large extent it was inspired by the study of rings of continuous functions in analysis and topology. I will mention rings of continuous functions from time to time, but will not attempt an in-depth discussion. With the rise of real algebraic geometry in the last 25 years there also arose a need for algebraic methods that are specifically adapted to real algebraic geometry, as opposed to general algebraic geometry. For a long time commutative algebra has been the answer to a similar need in general algebraic geometry. For real algebraic geometry, an important part of the answer is the theory of porings. However, the classical theory of porings frequently does not address the questions from real algebraic geometry. Therefore it is necessary to develop a more comprehensive theory of porings that includes rings of continuous functions, but also deals with phenomena occurring in real geometry. This is a major task that is not nearly finished at the present time. First I give the definition of porings: Definition 1 A partially ordered ring (poring) is a ring A (all rings are commutative with 1) together with a subset P ! A that satisfies the following conditions • P + P ! P ; • P ! P " P ; • A2 ! P ; • P ! "P = {0} .
    [Show full text]
  • Finitely Presented and Coherent Ordered Modules and Rings Friedrich Wehrung
    Finitely presented and coherent ordered modules and rings Friedrich Wehrung To cite this version: Friedrich Wehrung. Finitely presented and coherent ordered modules and rings. Communications in Algebra, Taylor & Francis, 1999, 27 (12), pp.5893–5919. 10.1080/00927879908826797. hal-00004049 HAL Id: hal-00004049 https://hal.archives-ouvertes.fr/hal-00004049 Submitted on 24 Jan 2005 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. FINITELY PRESENTED AND COHERENT ORDERED MODULES AND RINGS F. WEHRUNG Abstract. We extend the usual definition of coherence, for mod- ules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features: (i) Every subring of Q, and every totally ordered division ring, is po-coherent. (ii) For a partially ordered right module A over a po-coherent po- ring R, A is po-coherent if and only if A is a finitely presented R-module and A+ is a finitely generated R+-semimodule.
    [Show full text]
  • Formal Proofs in Real Algebraic Geometry: from Ordered Fields to Quantifier Elimination
    Logical Methods in Computer Science Vol. 8 (1:02) 2012, pp. 1{40 Submitted May 16, 2011 www.lmcs-online.org Published Feb. 16, 2012 FORMAL PROOFS IN REAL ALGEBRAIC GEOMETRY: FROM ORDERED FIELDS TO QUANTIFIER ELIMINATION CYRIL COHEN a AND ASSIA MAHBOUBI b a;b INRIA Saclay { ^Ile-de-France, LIX Ecole´ Polytechnique, INRIA Microsoft Research Joint Centre, e-mail address: [email protected], [email protected] Abstract. This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods. 1. Introduction Most interactive theorem provers benefit from libraries devoted to the properties of real numbers and of real valued analysis. Depending on the motivation of their developers these libraries adopt different choices for the definition of real numbers and for the material covered by the libraries: some systems favor axiomatic and classical real analysis, some other more effective versions.
    [Show full text]
  • Title Ordered Rings and Order-Preservation( Fulltext ) Author(S)
    Title Ordered rings and order-preservation( fulltext ) Author(s) KITAMURA,Yoshimi; TANAKA,Yoshio Citation 東京学芸大学紀要. 自然科学系, 64: 5-13 Issue Date 2012-09-28 URL http://hdl.handle.net/2309/131815 Publisher 東京学芸大学学術情報委員会 Rights Bulletin of Tokyo Gakugei University, Division of Natural Sciences, 64: 5 - 13，2012 Ordered rings and order-preservation Yoshimi KITAMURA* and Yoshio TANAKA* Department of Mathematics (Received for publication; May 25, 2012) KITAMURA, Y. and TANAKA, Y.: Ordered rings and order-preservation. Bull. Tokyo Gakugei Univ. Div. Nat. Sci., 64: 5-13 (2012) ISSN 1880-4330 Abstract We consider ordered rings or ordered fields, and give several related matters and examples. We give characterizations for residue class rings of ordered rings to be ordered rings (or ordered integral domains). Further, we give a characterization for an ordered ring R to satisfy the condition that all homomorphisms of R to any ordered ring are order-preserving. Key words and phrases: ordered ring, ordered field, residue class ring, order-preserving Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei-shi, Tokyo 184-8501, Japan Introduction The symbol R; Q; Z; or N is the field of real numbers; the field of rational numbers; the ring of integers; or the set of positive integers, respectively. The symbol R or R’ is a (non-zero) commutative ring, unless otherwise stated. We recall that R is called an ordered ring (or totally ordered ring) if it has a total order such that this ordering relation is order- preserving with respect to addition and multiplication. In particular, when R is a field, such an R is called an ordered field.
    [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]
  • Real Algebraic Geometry Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin
    Real Algebraic Geometry Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin To cite this version: Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin. Real Al- gebraic Geometry. Real Algberaic Geometry, Rennes, 2011, Jun 2011, Rennes, France. hal-00609687 HAL Id: hal-00609687 https://hal.archives-ouvertes.fr/hal-00609687 Submitted on 19 Jul 2011 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. Conference Real Algebraic Geometry Université de Rennes 1 20 - 24 june 2011 Beaulieu Campus Conference Real Algebraic Geometry Université de Rennes 1 20 - 24 june 2011 Beaulieu Campus Conference Real Algebraic Geometry C’est un très grand plaisir pour nous d’accueillir à Rennes cette conférence internationale de Géométrie Algébrique Réelle qui est le quatrième opus d’une belle série commencée il y a 30 ans. La toute première conférence internationale de la discipline fut organisée à Rennes en 1981 par Jean-Louis Colliot-Thélène, Michel Coste, Louis Mahé et Marie-Françoise Roy. Depuis, la discipline a pris un essor considérable : de très nombreuses conférences sur la thématique se sont déroulées dans le monde entier, un réseau européen a tissé des liens au cœur duquel Rennes 1 a toujours joué un rôle privilégié.
    [Show full text]