On the Complexity of Real Functions

Total Page:16

File Type:pdf, Size:1020Kb

On the Complexity of Real Functions On the Complexity of Real Functions Mark Braverman1 Department of Computer Science University of Toronto [email protected] Abstract with the computability notion of a single real number. To- day, we still do not have a unified and widely accepted no- We establish a new connection between the two most tion of real computability and complexity. common traditions in the theory of real computation, the Consider the two simplest objects over Rn: real sets and Blum-Shub-Smale model and the Computable Analysis ap- real functions. Several different approaches to the com- proach. We then use the connection to develop a notion putability of these objects have been proposed over the last of computability and complexity of functions over the re- seven decades. Unlike the discrete case, most approaches als that can be viewed as an extension of both models. We deal with computability of functions before the decidability argue that this notion is very natural when one tries to de- of sets. In the current paper we consider the two approaches termine just how “difficult” a certain function is for a very that are most common today: the tradition of Computable rich class of functions. Analysis – the “bit model”, and the Blum-Shub-Smale ap- proach. These two approaches have been developed to cap- ture different aspects of real computation, and in general are 1 Introduction not equivalent. The bit-model is very natural as a model for scientific computing when continuous functions on compact domains Questions of computability and complexity over the re- are√ involved. In particular, the “calculator” functions, such als are of extreme importance for understanding the connec- as x, sin x and log x are bit-computable, at least on some tions between nature and computations. Assessing the pos- properly chosen domains. None of these functions are com- sibilities of scientific computing in simulating and predict- putable in the BSS model, because computability in the BSS ing natural processes depends on an agreed upon and well model requires the function to have a very special algebraic studied notion of “real computation”. Addressing issues of structure (essentially being piecewise-rational). This makes physics and the Church-Turing Thesis also requires a strong the BSS notion inapplicable to scientific computing unless notion of computability over continuous spaces [23]. some modifications are made. The BSS model mimics some In the discrete setting, where the objects are from aspects of the actual way numerical analysts think about { }∗ 0, 1 , the concepts of computability and complexity are problems, and in fact [4] contains many strong results on very well studied. There are different approaches that have solving fundamental numerical problems in algebra, such been proved to yield equivalent definitions of computabil- as finding roots of a polynomial. The model’s disadvan- ity and (almost) equivalent definitions of complexity. From tage is the difficulty in interpreting negative results. The the Formal Logic point of view we have the notion of re- function ex and its graph are not BSS computable, and yet cursiveness, from Computational Complexity we have the we can easily compute the exponential function and plot its notion of a Turing Machine, and modeling the usual com- graph. Modifications that can be made to the model to deal puter yields the notion of abstract RAM. All these converge with these problems have been discussed in [20]. To our to the same well accepted concept of computability. knowledge, these modifications have not been formalized. In the continuous setting, where the objects are numbers On the other hand, there are very natural and simple R in , the situation is far less clear. Real Computation has functions that are BSS computable but not bit-computable. been studied since Turing’s original 1936 paper [21], where The simplest step function χ[0,∞) is an example. he introduced the Turing Machine. In that paper he dealt In the set setting, the models also often give fundamen- 1Research is partially supported by an NSERC postgraduate scholar- tally different answers to questions of computability. Con- ship sider, for example, the Koch snowflake K, the Julia set J = Jz2+1/4, and the Mandelbrot set M. In the BSS model The paper is organized as follows. In Section 2 we dis- none of these sets are computable (see [4]). In the bit model, cuss the bit model, and present some old and some new re- K is very easy to draw and is computable, J is computable, sults. In Section 3 we discuss the BSS model and possible although not as easily as K [3], and the question of whether modifications to it. In Section 4 we propose a new com- M is computable is open and depends on some deep con- putability and complexity definition for some discontinuous jectures from complex dynamics [14]. For most “common” and multi-valued functions. sets BSS is a more restrictive model, but this is not always the case, as demonstrated in section 3.2. 2TheBitModel We propose three simple natural modifications to the BSS model. Firstly, we allow the machines to use only com- putable constants. This amounts to working on a smaller 2.1 The Model of Computation field of the computable reals. Secondly, we allow the ma- chines to make some errors. This has been done infor- The computability of functions in the bit model as we mally before by analyzing the “error+condition number” of know it today was first proposed by Grzegorczyk [13] and a problem in the BSS model. Thirdly, we require an apriori Lacombe [16]. It has since been developed and generalized. estimate on the running time given as a function of the error More recent references on the subject include [15], [18], parameter ε. In section 3.2 we present an example show- and [22]. ing the importance of this condition. We show that under The basic model of computation here is an oracle Turing these modifications the BSS model becomes equivalent to D { k ∈ N ∈ N} Machine. Denote by = 2 : k , the the bit-model for sets. This involves simulating an infinite- set of the dyadic numbers. These are the rationals that have precision machine with a finite precision one. It can only be a finite binary expansion. An oracle for a number x is a done symbolically using the algebraic structure of the ma- function φ : N → D such that |φ(n)−x| < 2−n. The oracle chine’s constants and Q, followed by a general quantifier terminology is just a natural way to separate the complexity elimination over R. of computing x from the complexity of computing on x as a parameter. For most purposes one can think of the oracle φ for x as an infinite tape containing the binary expansion of x. Consider a function f : R → R. In plain language, a program M computing f would output a good approx- imation of f(x), provided approximations of the input x. More formally, the oracle machine M φ(n) outputs a 2−n- approximation of f(x) for any valid oracle φ for x.The definition extends naturally to a function f : Rk → R. φ1,φ2,...,φk Figure 1. The sets K (left), J (center), and M Here M = M (n) is allowed to query each of (right) the k parameters with an arbitrarily good precision and is required to output the values of f with precision 2−n. The primary goals of this paper are: 2.2 Basic Properties and Examples • to show that under the reasonable modifications men- tioned above, the two models become computationally One of the main properties of computable functions is that they are continuous. For a computable f : R → R, equivalent for sets; φ the machine Mf (n) can query the input x via φ with only • to propose a new notion of computability for functions finite level of precision before it outputs an approximation which extends both models, and takes advantage of q of f(x). This means that q should be a good approxima- some positive features in both; and tion of f(x) for all x in some small neighborhood of x. • based on the above, to propose the “right” notion of Thus f(x ) should be close to f(x) whenever x is close computational complexity for discontinuous functions, enough to x. A formal proof can be found in most standard extending naturally from the continuous case. references, e.g. in [22]. The computable ⇒ continuous property implies that This gives us the “right” notion of complexity for func- even the simplest discontinuous function, the step function tions for which no such notion√ previously existed. For ex- ample, the 2-valued function : C → C, as well as its 0 x<0 s0(x)= (1) single-valued branches defined on subsets of C. 1 x ≥ 0 2 is not computable under this definition. This is in contrary 1. If a pixel contains a point from S, then it is colored to the intuition that s0 must be a “simple” function. One black. This ensured that the entire set appears on the could also argue that some physical systems, e.g. quantum screen. energy levels, are best described using step functions and −n other simple discontinuous functions. One of the goals of 2. If a pixel is far (say 2 -far) from S, then it is colored the present work is to develop notions which deal with this white. This ensures that the picture is a faithful image problem. of S. We recall the classical definition of computable real We can take the pixels to be balls of radius 2−n with a numbers, introduced by Turing in [21].
Recommended publications
  • Bounded Pregeometries and Pairs of Fields
    BOUNDED PREGEOMETRIES AND PAIRS OF FIELDS ∗ LEONARDO ANGEL´ and LOU VAN DEN DRIES For Francisco Miraglia, on his 70th birthday Abstract A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields we introduce a certain kind of “bounded” pregeometry for such pairs. 1 Introduction The dimension of an algebraic variety equals the transcendence degree of its function field over the field of constants. We can use transcendence degree in a similar way to assign to definable sets in algebraically closed fields a dimension with good properties such as definable dependence on parameters. Section 1 contains a general setting for model-theoretic pregeometries and proves basic facts about the dimension function associated to such a pregeometry, including definable dependence on parameters if the pregeometry is bounded. One motivation for this came from the following issue concerning pairs (K, k) where K is an algebraically closed field and k is a proper algebraically closed subfield; we refer to such a pair (K, k)asa pair of algebraically closed fields, and we consider it in the usual way as an LU -structure, where LU is the language L of rings augmented by a unary relation symbol U. Let (K, k) be a pair of algebraically closed fields, and let S Kn be definable in (K, k). Then we have several notions of S being controlled⊆ by k, namely: S is internal to k, S is almost internal to k, and S is co-analyzable relative to k.
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • Sturm's Theorem
    Sturm's Theorem: determining the number of zeroes of polynomials in an open interval. Bachelor's thesis Eric Spreen University of Groningen [email protected] July 12, 2014 Supervisors: Prof. Dr. J. Top University of Groningen Dr. R. Dyer University of Groningen Abstract A review of the theory of polynomial rings and extension fields is presented, followed by an introduction on ordered, formally real, and real closed fields. This theory is then used to prove Sturm's Theorem, a classical result that enables one to find the number of roots of a polynomial that are contained within an open interval, simply by counting the number of sign changes in two sequences. This result can be extended to decide the existence of a root of a family of polynomials, by evaluating a set of polynomial equations, inequations and inequalities with integer coefficients. Contents 1 Introduction 2 2 Polynomials and Extensions 4 2.1 Polynomial rings . .4 2.2 Degree arithmetic . .6 2.3 Euclidean division algorithm . .6 2.3.1 Polynomial factors . .8 2.4 Field extensions . .9 2.4.1 Simple Field Extensions . 10 2.4.2 Dimensionality of an Extension . 12 2.4.3 Splitting Fields . 13 2.4.4 Galois Theory . 15 3 Real Closed Fields 17 3.1 Ordered and Formally Real Fields . 17 3.2 Real Closed Fields . 22 3.3 The Intermediate Value Theorem . 26 4 Sturm's Theorem 27 4.1 Variations in sign . 27 4.2 Systems of equations, inequations and inequalities . 32 4.3 Sturm's Theorem Parametrized . 33 4.3.1 Tarski's Principle .
    [Show full text]
  • Super-Real Fields—Totally Ordered Fields with Additional Structure, by H. Garth Dales and W. Hugh Woodin, Clarendon Press
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 91{98 S 0273-0979(98)00740-X Super-real fields|Totally ordered fields with additional structure, by H. Garth Dales and W. Hugh Woodin, Clarendon Press, Oxford, 1996, 357+xiii pp., $75.00, ISBN 0-19-853643-7 1. Automatic continuity The structure of the prime ideals in the ring C(X) of real-valued continuous functions on a topological space X has been studied intensively, notably in the early work of Kohls [6] and in the classic monograph by Gilman and Jerison [4]. One can come to this in any number of ways, but one of the more striking motiva- tions for this type of investigation comes from Kaplansky’s study of algebra norms on C(X), which he rephrased as the problem of “automatic continuity” of homo- morphisms; this will be recalled below. The present book is concerned with some new and very substantial ideas relating to the classification of these prime ideals and the associated rings, which bear on Kaplansky’s problem, though the precise connection of this work with Kaplansky’s problem remains largely at the level of conjecture, and as a result it may appeal more to those already in possession of some of the technology to make further progress – notably, set theorists and set theoretic topologists and perhaps a certain breed of model theorist – than to the audience of analysts toward whom it appears to be aimed. This book does not supplant either [4] or [2] (etc.); it follows up on them.
    [Show full text]
  • Pseudo Real Closed Field, Pseudo P-Adically Closed Fields and NTP2
    Pseudo real closed fields, pseudo p-adically closed fields and NTP2 Samaria Montenegro∗ Université Paris Diderot-Paris 7† Abstract The main result of this paper is a positive answer to the Conjecture 5.1 of [15] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then T h(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then T h(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then T h(M) is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking. Keywords: Model theory, ordered fields, p-adic valuation, real closed fields, p-adically closed fields, PRC, PpC, NIP, NTP2. Mathematics Subject Classification: Primary 03C45, 03C60; Secondary 03C64, 12L12. Contents 1 Introduction 2 2 Preliminaries on pseudo real closed fields 4 2.1 Orderedfields .................................... 5 2.2 Pseudorealclosedfields . .. .. .... 5 2.3 The theory of PRC fields with n orderings ..................... 6 arXiv:1411.7654v2 [math.LO] 27 Sep 2016 3 Bounded pseudo real closed fields 7 3.1 Density theorem for PRC bounded fields . ...... 8 3.1.1 Density theorem for one variable definable sets . ......... 9 3.1.2 Density theorem for several variable definable sets. ........... 12 3.2 Amalgamation theorems for PRC bounded fields . ........ 14 ∗[email protected]; present address: Universidad de los Andes †Partially supported by ValCoMo (ANR-13-BS01-0006) and the Universidad de Costa Rica.
    [Show full text]
  • Factorization in Generalized Power Series
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 2, Pages 553{577 S 0002-9947(99)02172-8 Article electronically published on May 20, 1999 FACTORIZATION IN GENERALIZED POWER SERIES ALESSANDRO BERARDUCCI Abstract. The field of generalized power series with real coefficients and ex- ponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the 0 ring R((G≤ )) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given 1=n by Conway (1976): n t− + 1. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible.P We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G =(R;+;0; ) we can give the following test for irreducibility based only on the order type≤ of the support of the series: if the order type α is either ! or of the form !! and the series is not divisible by any mono- mial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case G = R. In the final part of the paper we study the irreducibility of series with finite support. 1. Introduction 1.1. Fields of generalized power series. Generalized power series with expo- nents in an arbitrary abelian ordered group are a classical tool in the study of valued fields and ordered fields [Hahn 07, MacLane 39, Kaplansky 42, Fuchs 63, Ribenboim 68, Ribenboim 92].
    [Show full text]
  • Real Closed Fields
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1968 Real closed fields Yean-mei Wang Chou The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Chou, Yean-mei Wang, "Real closed fields" (1968). Graduate Student Theses, Dissertations, & Professional Papers. 8192. https://scholarworks.umt.edu/etd/8192 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. EEAL CLOSED FIELDS By Yean-mei Wang Chou B.A., National Taiwan University, l96l B.A., University of Oregon, 19^5 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MONTANA 1968 Approved by: Chairman, Board of Examiners raduate School Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38993 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OwMTtation PVblmhing UMI EP38993 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author.
    [Show full text]
  • On the Topological Aspects of the Theory of Represented Spaces∗
    On the topological aspects of the theory of represented spaces∗ Arno Pauly Clare College University of Cambridge, United Kingdom [email protected] Represented spaces form the general setting for the study of computability derived from Turing ma- chines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by ESCARDO´ ,SCHRODER¨ , and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property. 1 Introduction ∗ Just as numberings provide a tool to transfer computability from N (or f0;1g ) to all sorts of countable structures; representations provide a means to introduce computability on structures of the cardinality of the continuum based on computability on Cantor space f0;1gN. This is of course essential for com- n m k n m putable analysis [66], dealing with spaces such as R, C (R ;R ), C (R ;R ) or general Hilbert spaces [8]. Computable measure theory (e.g. [65, 58, 44, 30, 21]), or computability on the set of countable ordinals [39, 48] likewise rely on representations as foundation. Essentially, by equipping a set with a representation, we arrive at a represented space – to some extent1 the most general structure carrying a notion of computability that is derived from Turing ma- chines2.
    [Show full text]
  • Computability and Analysis: the Legacy of Alan Turing
    Computability and analysis: the legacy of Alan Turing Jeremy Avigad Departments of Philosophy and Mathematical Sciences Carnegie Mellon University, Pittsburgh, USA [email protected] Vasco Brattka Faculty of Computer Science Universit¨at der Bundeswehr M¨unchen, Germany and Department of Mathematics and Applied Mathematics University of Cape Town, South Africa and Isaac Newton Institute for Mathematical Sciences Cambridge, United Kingdom [email protected] October 31, 2012 1 Introduction For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems,” which assert that a construction can be carried out to meet a given specification, and “theorems,” which assert that some property holds of a particular geometric configuration. For example, Proposition 10 of Book I reads “To bisect a given straight line.” Euclid’s “proof” gives the construction, and ends with the (Greek equivalent of) Q.E.F., for quod erat faciendum, or “that which was to be done.” arXiv:1206.3431v2 [math.LO] 30 Oct 2012 Proofs of theorems, in contrast, end with Q.E.D., for quod erat demonstran- dum, or “that which was to be shown”; but even these typically involve the construction of auxiliary geometric objects in order to verify the claim. Similarly, algebra was devoted to developing algorithms for solving equa- tions. This outlook characterized the subject from its origins in ancient Egypt and Babylon, through the ninth century work of al-Khwarizmi, to the solutions to the quadratic and cubic equations in Cardano’s Ars Magna of 1545, and to Lagrange’s study of the quintic in his R´eflexions sur la r´esolution alg´ebrique des ´equations of 1770.
    [Show full text]
  • From Constructive Mathematics to Computable Analysis Via the Realizability Interpretation
    From Constructive Mathematics to Computable Analysis via the Realizability Interpretation Vom Fachbereich Mathematik der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Math. Peter Lietz aus Mainz Referent: Prof. Dr. Thomas Streicher Korreferent: Dr. Alex Simpson Tag der Einreichung: 22. Januar 2004 Tag der m¨undlichen Pr¨ufung: 11. Februar 2004 Darmstadt 2004 D17 Hiermit versichere ich, dass ich diese Dissertation selbst¨andig verfasst und nur die angegebenen Hilfsmittel verwendet habe. Peter Lietz Abstract Constructive mathematics is mathematics without the use of the principle of the excluded middle. There exists a wide array of models of constructive logic. One particular interpretation of constructive mathematics is the realizability interpreta- tion. It is utilized as a metamathematical tool in order to derive admissible rules of deduction for systems of constructive logic or to demonstrate the equiconsistency of extensions of constructive logic. In this thesis, we employ various realizability mod- els in order to logically separate several statements about continuity in constructive mathematics. A trademark of some constructive formalisms is predicativity. Predicative logic does not allow the definition of a set by quantifying over a collection of sets that the set to be defined is a member of. Starting from realizability models over a typed version of partial combinatory algebras we are able to show that the ensuing models provide the features necessary in order to interpret impredicative logics and type theories if and only if the underlying typed partial combinatory algebra is equivalent to an untyped pca. It is an ongoing theme in this thesis to switch between the worlds of classical and constructive mathematics and to try and use constructive logic as a method in order to obtain results of interest also for the classically minded mathematician.
    [Show full text]
  • Perron-Frobenlus Theory Over Real Closed Fields and Fractional Power Series Expanslons*
    I NOImI-HOILAND Perron-Frobenlus Theory Over Real Closed Fields and Fractional Power Series Expanslons* B. Curtis Eaves Department of Operations Research Stanford University Stanford, California 94305-4022 Uriel G. Rothblum Faculty of Industrial Engineering and Management Technion-Israel Institute of Technology Haifa 32000, Israel and Hans Schneider Department of Mathematics University of Wisconsin Madison, Wisconsin 53706 Submitted by Raphael Loewy ABSTRACT Some of the main results of the Perron-Frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field. We use the results to prove the existence of a fractional power series expansion for the Perron-Frobenius eigenvalue and normalized eigenvector of real, square, nonnegative, irreducible matrices which are obtained by perturbing a (possibly reducible) nonnega­ tive matrix. Further, we identify a system of equations and inequalities whose solution 'Research was supported in part by National Science Foundation grants DMS-92-07409, DMS-88-0144S and DMS-91-2331B and by United States-Israel Binational Science Foundation grant 90-00434. LINEAR ALGEBRA AND ITS APPLICATIONS 220:123-150 (995) © Elsevier Science Inc., 1995 0024-3795/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0024-3795(94)OOO53-G 124 B. C. EAVES, U. G. ROTHBLUM, AND H. SCHNEIDER yields the coefficients of these expansions. For irreducible matrices, our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence. 1. INTRODUCTION In this paper we obtain convergent fractional power series expansions for the Perron-Frobenius eigenvalue and normalized eigenvector for real, square, nonnegative, irreducible matrices which are obtained by perturbing (not necessarily irreducible) nonnegative matrices.
    [Show full text]
  • Using Infinitesimals for Algorithms in Real Algebraic Geometry
    Using infinitesimals for algorithms in real algebraic geometry by Marie-Françoise Roy IRMAR (UMR CNRS 6625), Université de Rennes IPAM,April 12, 2014 talk based on several papers written with S. Basu and/or R. Pollack see Algorithms in Real Algebraic Geometry S. Basu, R. Pollack, M.-F. R. 1 2 Section 1 1 Motivation critical point method : basis for many quantitative results in real algebraic geometry (starting with Oleinik-Petrovskii-Thom-Milnor) wish : use critical point method in algorithms in real algebraic geometry with singly exponential complexity in problems such as existential theory of the reals, deciding connectivity (and more) also in the quadratic and symmetric case (see talk by Cordian Riener) requires general position, reached through various deformation tricks in these algorithms, "small enough" is through the use of infinitesimals Motivation 3 Cylindrical Decomposition is doubly exponential (unavoidable if use repeated projections, see preceeding lectures) Critical point method based on Morse theory nonsingular bounded compact hypersurface V = M Rk , H(M) = 0 , { ∈ } i.e. such that ∂H ∂H GradM(H)= (M),..., (M) ∂X1 ∂Xk does not vanish on the zeros of H in Ck critical points of the projection on the X1 axis meet all the connected components of V − 4 Section 1 k−1 when the X1 direction is a Morse function, there are d(d 1) such critical points (Bezout), − ∂H ∂H H(M)= (M)= ..., (M) = 0, (1) ∂X2 ∂Xk (project directly on one line rather than repeated projections) so : want to find a point in every connected components in time polynomial in (1/2)d(d 1)k−1 i.e.
    [Show full text]