Informatique Mathématique une photographie en X et Y et Z et... (Eds.)

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1 Informatique Mathématique une photographie en 2015 X et Y et Z et... (Eds.) 9 mars 2015

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3 Table des matières 1 Algorithms in real algebraic geometry Introduction State of the art and purpose of this course Basic objects : algorithms and degree bounds Basic definitions Critical points, values and their properties Transversality results and properties of critical points and values Generalized critical values Polynomial System Solving : the zero-dimensional case Degree bounds and computation of critical points Computing sample points in real algebraic sets Reduction to the algebraic case The smooth equidimensional case Deformation techniques and singular algebraic sets Connectivity in real algebraic sets Definitions, data-representation and use of the output Practical improvement of Canny s algorithm A baby steps/giant steps approach for smooth bounded hypersurfaces Towards quantifier elimination Polynomial Optimization Variant real quantifier elimination

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5 Table des figures 1.1 Critical points associated to projections of the sphere Example f = X 1 (X 1 X 2 1) Example f = X1 2 + (X 1X 2 1) Change of coordinates for computing sample points Roadmaps using Canny s algorithm Example of divide-and-conquer strategy to compute roadmaps Example for the Variant Quantifier Elimination Algorithm. 47

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7 Liste des tableaux

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9 Chapitre 1 Efficient algorithms in real algebraic geometry Pierre Lairez Mohab Safey El Din Solving non-linear algebraic problems is one of the major challenges in scientific computing. In several areas of engineering sciences, algebraic problems encode geometric conditions on variables taking their values over the reals. Thus, most of the time, one aims to obtain some informations on the real solution set of polynomial systems. The resolution of these problems has often a complexity which is singly exponential in the number of variables while most of the software implement algorithms which are doubly exponential in the number of variables. Moreover, the non-linear nature of the considered algebraic problems and the necessity to ensure the quality of the results of our computations leads to use Computer Algebra techniques to handle these problems. In this context, the issues are to develop exact algorithms, to control their complexity suitably, to implement efficient software and to identify useful specifications for the end-users. In this course, we present recent geometric ideas that lead to design algorithms whose complexities are singly exponential in the number of variables, at least in generic situations, and whose practical behaviours reflect the complexity gains which are obtained. We focus on the computation of real solutions to multivariate polynomial systems, connectivity queries and simple quantifier elimination problems.

10 4 Chapitre 1. Algorithms in real algebraic geometry 1.1 Introduction Let f 1,..., f p, g 1,..., g s be polynomials in Q[X 1,..., X n ] and let S R n be the semi-algebraic set defined by f 1 = = f p = 0, g 1 > 0,..., g s > 0. (1.1) Such polynomial systems appear in many fields of engineering sciences, especially those where geometric constraints (distance between sets of points, rank defects of matrices with polynomial entries and/or positivity of some of their minors, etc.) govern the objects under study. Naturally, the meaning of solving such a polynomial system depends on the geometric information which is expected by the end-user. We focus in this course on three specifications : Computing sample points : Given (1.1) decide the emptiness of S and/or compute at least one point in each connected component of S. This problem has many applications, e.g. in robotics, computational geometry, pattern-matching, signal theory, camera calibration, etc. Connectivity queries : Given (1.1) and a finite set of points P S, decide which elements of P lie in the same connected components of S. This problem is solved by computing a roadmap for S and P, i.e. a curve containing P, contained in S and that has a non-empty and connected intersection with every connected component of S. Motion planning problems can be solved by answering to such connectivity queries [36, 9]. It also arises in classification problems in computational geometry. Quantifier elimination : Given (1.1) and a projection π : R n E where E is a linear affine subspace of R n, describe π(s). This problem, seen as a quantifier elimination problem, arises in many fields such as global optimization, stability analysis in biology, numerical schemes for PDEs and initial boundary value problems, program verification and automated theorem proving, etc. In addition to their direct interests, algorithms solving the three aforementioned problems can be used as subroutines in other algorithms dealing with real solutions of polynomial systems : Solving systems with parameters (zero-dimensional case). Consider a polynomial system S depending on r parameters and suppose it has a finite number of complex solutions for generic specializations of the parameters. Determining the number of real solutions of S (according to the parameters values) is solved by constructing a real algebraic variety D R r in the parameter space, such

11 1.1. Introduction 5 that the number of real roots of S is invariant over each connected component of R r \ D. Thus, in order to complete the resolution of such systems, it suffices to compute at least one point in each connected component of R r \ D. Therefore, algorithms for computing efficiently sample points in semi-algebraic sets may play a crucial role in the solving process of parametric systems. Description of the connected components of S. The semi-algebraic set S is the union of finitely many connected components. Each connected component can be described as the solution set of a system of polynomial equations and inequalities. For example, if S is the hyperbola defined by X 1 X 2 1 = 0, one connected component can be described by X 1 X 2 1 = 0 and X 1 > 0 and the other one can be described by X 1 X 2 1 = 0 and X 1 < 0. In [5, Chapter 16], algorithms for solving connectivity queries by means of roadmap computations are used to derive an algorithm for computing semi-algebraic descriptions of the connected components of S State of the art and purpose of this course As above, given f 1,..., f p, g 1,..., g s in Q[X 1,..., X n ], S R n denotes the semi-algebraic set defined by f 1 = = f p = 0, g 1 > 0,..., g s > 0. We also consider the algebraic variety V C n defined by f 1 = = f p = 0. In the sequel, D = max(deg(f i ), deg(g j ), 1 i p, 1 j s). We adopt the classical arithmetic complexity model over Q : all basic field operations and comparisons of elements are counted at unit cost. We denote by M(D) the cost of multiplying two polynomials in Q[X] of degree at most D and ω denotes the linear algebra exponent. Deciding the emptiness and/or computing sample points in semialgebraic sets. We may make a distinction between the problem of deciding the emptiness of S and the problem of computing sample points in each connected component of S. In the latter case, the computational cost is at least equal to the number of connected components of S. This number is itself upper bounded by Thom-Milnor s bound, i.e. s n O(D) n. This bound is singly exponential in n. For some families of semi-algebraic sets, the number of connected components does vary exponentially in n. Therefore, any general algorithm for computing at least one point in each connected component of a semi-algebraic set must have a complexity at least singly exponential in n. Deciding the emptiness of S can be also done by computing algebraic certificates related to the Positivstellensatz

12 6 Chapitre 1. Algorithms in real algebraic geometry [6, Chapter 4]. For example, proving that a polynomial f Q[X 1,..., X n ] is positive over R n can be done by writing f as a sum of squares in Q(X 1,..., X n ) (see [1]). In this course, we will focus on algorithms that compute explicitly sample points in semi-algebraic sets ; in other words, we will not consider algorithms based on computing algebraic certificates of infeasibility as the Positivstellensatz. Computing sample points in semi-algebraic sets can be done using Collins Cylindrical Algebraic Decomposition Algorithm [10]. This algorithm allows to partition semi-algebraic sets in cylindrical cells which are homeomorphic to R i (for some i N). To do that, variables that arise in the polynomial family defining the semi-algebraic sets are eliminated one after another using a recursive projection operator based on subresultant sequences. Thus the quadratic degree growth induced by these subresultant computations at each recursive call leads to a complexity ((p + s)d) O(1)n, which is doubly exponential in n. Cylindrical Algebraic Decomposition has been intensively studied and improved. However, in practice, the best implementations are usually limited to problems involving a number of variables less than or equal to 4 because of the doubly exponential complexity of the algorithm. One could expect a complexity which is polynomial in Thom- Milnor s bound. Such a result has been obtained in [19] with several improvements (see [5] and references therein). These algorithms rely on the so-called critical point method. It is based on the basic fact that any polynomial mapping reaches its extrema at critical points on each connected component of a compact real algebraic set. So the idea is to choose a polynomial mapping reaching its extrema at each connected component of the studied real algebraic set. Those points at which these extrema are reached are among those points that we call critical points. In addition to that, one might expect that these critical points are solutions of a zero dimensional system of polynomial equations (i.e. with a finite number of complex solutions) whose associated Bézout-bound is singly exponential in the number of variables. Indeed, in this situation, one can use algebraic tools (such as Gröbner bases) to compute rational parametrizations of the solutions and hence isolate the real ones. This process is extended to semi-algebraic sets by a reduction from the general case of semi-algebraic sets to the study of a real solution set, that is smooth and bounded, of a single polynomial equation. Algebraic manipulations (sums-of-squares of input polynomials) and several infinitesimal deformations are performed to enable such reductions. The outcome is a deterministic algorithm whose complexity is s n D O(n) (in this complexity s n is a combinatorial factor, while D O(n) is an algebraic factor)

13 1.1. Introduction 7 which is much better than the doubly exponential running time of Cylindrical Algebraic Decomposition. These algorithms based on the critical point method do not lead to practical performances which reflect the theoretical complexity breakthrough. Therefore, it is a relevant question to ask whether one can derive algorithms, inspired by the critical point method, that run in time D O(n) and whose practical behaviour reflect the complexity gain. Connectivity queries in semi-algebraic sets. In [36], Schwartz and Sharir exhibit the interest of being able to answer connectivity queries in semi-algebraic sets. The application domain is motion planning : consider a rigid body (or a partly rigid body), that can move according to some geometric transformations (translations, rotations) in a space containing some obstacles. The issue is to know if the body can move from one configuration to another without meeting the obstacles. Considering the configuration space of the problem, one can answer such questions by deciding if some points lie in the same connected component of a given semialgebraic set. A well-known instance of such a problem is the Piano Mover s Problem [36]. Schwartz and Sharir have designed in [36] an algorithm based on Cylindrical Algebraic Decomposition to answer connectivity queries in semi-algebraic sets. Thus, the complexity of this algorithm is at least doubly exponential in the number of variables. This is a major problem because even if we consider motion planning problems in R 3, the corresponding configuration space can lie in R n with n 3 since one has to take into account all the degrees of freedom on the possible geometric transformations. In [9], Canny introduces the notion of roadmap. The idea is to reduce the problem of answering a connectivity query in a semi-algebraic set of arbitrary dimension to the one dimensional case : for semi-algebraic curves, one can use efficient algorithms answering connectivity queries. Thus, given a finite set of points P S, a roadmap of (S, P) is a semialgebraic curve containing P and having a non-empty and connected intersection with each connected component of S. Canny gave a probabilistic algorithm running in time ((p+s)d) O(n2) [9]. This algorithm also relies on the computation of critical points of some polynomial mappings. Several improvements were brought to Canny s method (see [5] and references therein). This series of works culminates with the deterministic algorithm given in [4] running in time (p + s) d D O(n2) (where d = dim(v R n )). It produces a semi-algebraic path connecting the points which lie in the same connected component. Nevertheless, for similar reasons to the

14 8 Chapitre 1. Algorithms in real algebraic geometry ones described in the previous paragraph, the practical behavior of these latter algorithms does not reflect the complexity breakthrough. Hence, for this problem, it is relevant to ask whether one can obtain algorithms that have a satisfactory behaviour, i.e. that can tackle problems that are out of reach of the best implementations of the Cylindrical Algebraic Decomposition algorithm. It is also relevant to ask if one can get algorithms with a better complexity than D O(n2), e.g. algorithms that run in time D O(nα) with α < 2. Quantifier elimination. Consider a quantified formula Φ : (X 1,..., X n ) R n f 1 = = f p = 0, g 1 > 0,..., g s > 0 where the f i s and the g i s are polynomials in Q[X 1,..., X n, Y 1,..., Y r ]. The variables Y 1,..., Y r are called free variables (or parameters), the other ones are called quantified variables. In [38], Tarski proved that one can decide the emptiness of S = {y R r Φ(X, y) is true } and provide a formula defining S. This also allows to consider formulas with more than one block of variables. The complexity of Tarski s algorithm was not elementary recursive. A variant of Cylindrical Algebraic Decomposition [10] allows to solve such quantifier elimination problems, still with a doubly exponential complexity in the total number of variables and parameters and a huge amount of work has been done in this direction. In [19], Grigoriev gave an algorithm for deciding if a quantified formula is true taking into account the block structure : the given algorithm has a complexity doubly exponential in the number of blocks of variables. This has been extended to general quantifier elimination problems (see [5] and references therein). These algorithms also use infinitesimal deformations and similar algebraic manipulations such as sums-of-squares of the input polynomials, etc. Again, the practical behavior does not reflect the theoretical complexity breakthrough. Also, the best current implementations of Cylindrical Algebraic Decomposition can only handle problems involving at most four variables. Again, one may expect to obtain an algorithm whose complexity lie in the best known complexity class but whose practical behaviour reflects the complexity gain. Content of the course All in all, the above state of the art shows that there is a gap to fill between algorithms used in practice and algorithms based on the critical point method that have a good theoretical complexity. During the last

15 1.2. Basic objects : algorithms and degree bounds 9 decade, tremendous efforts have been made to design algorithms whose complexities match the best known bounds (or sometimes improve them) and whose practical behaviours reflect the complexity gains that are obtained (at least compared to the Cylindrical Algebraic Decomposition algorithm whose complexity is doubly exponential in the number of variables). These algorithms are still based on computing critical points. The approaches that are present here try to exploit topological and geometrical properties of critical loci of projections on generic lines (or generic affine subspaces) to speed up the computations required for solving the aforementioned algorithmic problems. Since we use randomization, complexity analyzes are done in the probabilistic framework of the geometric resolution algorithm (as in [2]) to ensure that, under genericity assumptions, our algorithms have a complexity which almost match the best complexity bounds given in [5]. This will also allow us to make explicit the complexity constants in the exponent. In practice, our implementations rely on deterministic algorithms and use Gröbner bases which still provide nowadays the most efficient way to solve zero dimensional systems in general. Acknowledgments. Most of the material described in this course is a simplified version of joint research articles with our collaborators. In particular, we thank Bernd Bank, Jean-Charles Faugère, Marc Giusti, Joos Heintz, Hoon Hong, Eric Schost and Pierre-Jean Spaenlehauer among many others. 1.2 Basic objects : algorithms and degree bounds As explained in Section 1.1, the algorithms that are described further are based on subroutines for isolating the real solutions of polynomial systems that have finitely many complex solutions. The first subsection is dedicated to recall basic definitions and elementary properties of the complex solution sets of polynomial systems of equations with complex coefficients. These sets are called algebraic sets or algebraic varieties and the mathematical area dealing with such objects (and maps between them) is algebraic geometry. The mathematical area called real algebraic geometry focuses on studying the real solution sets of polynomial systems of equations and inequalities with real coefficients (and maps between them) ; these sets are called real algebraic sets when the considered system contains only equations, else they are called semi-algebraic sets.

16 10 Chapitre 1. Algorithms in real algebraic geometry Real algebraic sets and semi-algebraic sets have nice properties. In particular, their number of connected components is finite and semialgebraic sets are stable by projection. We refer to [6] for basic properties on real and semi-algebraic sets. In the next subsection, we recall the definition of critical points and values of the restriction of a polynomial map to the complex solution set of a polynomial system of equations with coefficients in C. These definitions extend easily to the real case. We also recall classical transversality results that are used to prove several of the algorithms given later for studying real algebraic sets. The next subsection is dedicated to generalized critical values. This is a more recent notion which extends the notion of critical values. Finally, we overview symbolic computation techniques that allow to rewrite the input system in a suitable form for real root extraction. When the input system defines a curve, we will use these routines to rewrite the input in a similar form. This section ends with a short overview of these algorithms. We mainly focus on their inputs and outputs and discuss their complexities. The content of the next subsections is basic ; we provide many references for the interested readers Basic definitions Algebraic sets and Zariski topology. Let K be a field containing Q. A subset V C n is a K-algebraic set (or K-algebraic variety) if there exist (f 1,..., f p ) K[X 1,..., X n ] such that V is the solution set in C n of the system f 1 = = f p = 0. In this course, we will mostly consider Q-algebraic varieties that are simply called algebraic varieties. The Zariski topology on C n is a topology where the closed sets are the algebraic varieties. Given a set U C n, the Zariski closure of U is the smallest algebraic set containing U. It is denoted by U Z. A Zariski open set is the complement of a Zariski closed set. A finite intersection of Zariski closed and open sets is a constructible set. Sometimes, we will say that a property, depending on some parameters, is generic ; this will mean that in this parameters space, there exists a non-empty Zariski open set such that for any specialization of the parameters in this set, the property is satisfied. For instance, we will say that a property is true in generic coordinates if there exists a non-empty Zariski

17 1.2. Basic objects : algorithms and degree bounds 11 open set O in GL n (C) such that for any A O, the property is satisfied after performing the change of coordinates x Ax. An algebraic variety V is reducible if it can be written as the union of two proper algebraic varieties, else it is irreducible. For any variety V, there exist irreducible varieties V 1,..., V s such that for i j, V i V j and such that V = V 1 V s. The algebraic varieties V i are the irreducible components of V. The decomposition of V as the union of its irreducible components is unique. The dimension of an algebraic set V is the largest integer d such that there exists {i 1,..., i d } for which the projection of V on X i1,..., X id has non-empty interior. By convention, the dimension of the empty set is 1. Algebraic sets of dimension 0 are non-empty and finite. We write dim(v ) = d. The variety is equidimensional of dimension d if all its irreducible components have dimension d. Finally, let W be a constructible set. We define its dimension as the dimension of its Zariski closure. Ideals and varieties. Hence, an algebraic set V C n will be given by a polynomial system f 1 = = f p = 0 in Q[X 1,..., X n ]. The ideal generated by (f 1,..., f p ) and denoted by f 1,..., f p is the set of polynomial that can be written as f = q 1 f q p f p with q i Q[X 1,..., X n ]. Note that for any f f 1,..., f p and any x V, f(x) = 0. Hence, given an ideal, it makes sense to speak about the algebraic set associated to this ideal. One can easily prove that the set of polynomials that vanish on V is also an ideal ; we will denote it by I(V ). Note that I(V ) may not be equal to f 1,..., f p. Indeed, consider the algebraic set V defined by X 2 1 = 0 and remark that I(V ) = X 1. This leads to consider the following. Given an ideal I Q[X 1,..., X n ], one can again prove easily that {f Q[X 1,..., X n ] k Nf k I} is an ideal ; it is called the radical ideal of I. An ideal will be said to be radical iff it equals its radical. Further, we will consider the saturation of an ideal I by an ideal J : it is the ideal {f Q[X 1,..., X n ] g J k N fg k I} ; we will denote it by I : J. The algebraic set associated to I : J is the Zariski closure of the set difference V (I) \ V (J). Regular and singular points. Let V C n be an algebraic variety. We denote by I(V ) the set of all polynomials f that vanish on V. The ideal

18 12 Chapitre 1. Algorithms in real algebraic geometry I(V ) is radical. The Zariski tangent space to V at x V is the vector space defined by the equations f X 1 (x)v f X n (x)v n = 0 for all f I(V ) ; this vector space will be denoted by T x V in the sequel. Let F = (f 1,..., f p ) Q[X 1,..., X n ] be a set of generators of I(V ). This means that for any f I(V ), there exists (q 1,..., q p ) Q[X 1,..., X n] such that f = q 1 f q p f p. Then, the Zariski tangent space to V at x is the right kernel of the jacobian matrix associated to F f 1 X 1 jac(f) =... f p X 1 f 1 X n. f p X n evaluated at x. Suppose that V is equidimensional. The regular points on V are those points x V such that dim(t x V ) = dim(v ). By the Jacobian criterion [13, Theorem 16.19], the regular points are those points at which the rank of jac(f) evaluated at x is n dim(v ). The singular points are those points on V which are not regular. The set of singular points of V is an algebraic set. Indeed, it is defined by the vanishing of the polynomials in F and the (n dim(v )+1, n dim(v )+1) minors of jac(v ). The set of regular (resp. singular) points of an algebraic variety V will be denoted by reg(v ) (resp. sing(v )). An equidimensional algebraic set V C n such that sing(v ) = is said to be smooth. If F = (f 1,..., f p ) Q[X 1,..., X n ] is a set of generators of I(V ), then there is no point in V at which jac(f) has rank less than n dim(v ). Properness of a map. We will use the notion of properness of a map, which we now introduce, together with the notion of a dominant map. Let ϕ : V W be a map of topological spaces. The map ϕ is proper at w W if there is a neighborhood B of w such that ϕ 1 (B) is compact, where B denotes the closure of B. We will consider maps between complex or real algebraic varieties. The notion of properness will be relative to the topologies induced by the metric topologies of C or R. It is easy to prove that the image of a closed set by a proper map is closed. Next, a map of irreducible complex varieties ϕ : V W is dominant if its image is dense in W, i.e. if the dimension of ϕ(v ) as a complex constructible set equals the dimension of W. We extend this definition to the case

19 1.2. Basic objects : algorithms and degree bounds 13 of a map V W, where V is not necessarily irreducible by requiring that the restriction of ϕ to each irreducible component of V is dominant. Let now V C n be an algebraic variety of dimension d and π : (x 1,..., x n ) V (x 1,..., x d ) C d be a dominant projection (note that this implies that all irreducible components of V have dimension d). Then by the theorem of dimension of fibers [37, Chapter 1.6], π has generically finite fibers. Indeed, this theorem tells us that for α generically chosen in π(v ), the following holds : dim(v ) = dim(π(v )) + dim(π 1 (α)) In this situation, the set of points of C d at which π is not proper is a hypersurface [22]. In [22, Lemma 3.10], Jelonek provides an algebraic characterization of the set of non-properness of the restriction of π to V. An algorithm, based on this characterization, for computing the set of non-properness of a dominant mapping restricted to an equidimensional algebraic variety V is designed in [32]. We denote by NonProperness an algorithm which takes as input a polynomial family F and a list of variables [X 1,..., X d ] and which computes a polynomial whose zero-set is the set of non-properness of π. Note that this implies that any assumption based on the properness of a given projection restricted to an algebraic set can be checked computationally. This will be used in the algorithms described in the next sections Critical points, values and their properties Hereafter, we consider an equidimensional algebraic set V C n. Critical points, polar varieties and critical values. Now, let ϕ : x C n (ϕ 1 (x),..., ϕ k (x)) with ϕ i Q[X 1,..., X n ]. The set of critical points of the restriction of ϕ to V is the set of points x reg(v ) such that d x ϕ(t x V ) C i ; we will denote it by crit(ϕ, reg(v )). A critical value of the restriction of ϕ to V is a point in the image of crit(ϕ, reg(v )) by ϕ. A point y C k is said to be a regular value of the map ϕ if it is neither a critical value nor in ϕ(sing(v )). The polar variety associated to the restriction of ϕ to V is the Zariski closure of crit(ϕ, reg(v )) ; we will denote it by W(ϕ, V ). Sometimes, it will be convenient to consider crit(ϕ, V ) = crit(ϕ, reg(v )) sing(v ).

20 14 Chapitre 1. Algorithms in real algebraic geometry x 3 x 1 x 2 FIGURE 1.1 Critical points associated to projections of the sphere Algebraic characterization of critical points and polar varieties. sequel, we denote by jac([f 1,..., f p ], X i ) the jacobian matrix f 1 X i... f p X i f 1 X n. f p X n In the If i = 1, jac(f 1,..., f p ) denotes jac([f 1,..., f p ], [X 1,..., X n ]). If J is a matrix with entries in a ring, MaxMinors(J) denotes the set of maximal minors of J. Suppose first that f 1,..., f p is radical and equidimensional of codimension p. Then, crit(ϕ, V ) is defined by the family f 1,..., f p, MaxMinors(jac([f 1,..., f p, ϕ 1,..., ϕ k ])). (1.2) Note that (1.2) can be overdetermined (and it will be as soon as p 2 and k = 1). Obviously, if V is smooth, (1.2) defines also crit(ϕ, reg(v )). Denoting by I the ideal generated by (1.2), the ideal defines W(ϕ, V ). I : MaxMinors(jac([f 1,..., f p ]))

21 1.2. Basic objects : algorithms and degree bounds 15 Let us give an example that is illustrated by Figure 1.1. Consider the algebraic set defined by the equation X X (X 3 1) 2 1 = 0 and the projections π i : (x 1,..., x n ) (x 1,..., x i ) for i = 1 and 2. Note that it is smooth because there is no point at which all the partial derivatives of X X2 2 + (X 3 1) 2 1 vanish. The real trace of V is the sphere. The polar variety crit(π 2, V ) associated to π 2 (hence the projection on the (X 1, X 2 )-plane) is then defined by the system of equation X X (X 3 1) 2 1 = X 3 1 = 0. Its real trace is the red curve that is the equator on the sphere. The polar variety crit(π 1, V ) associated to π 1 and V is defined by the system of equations X X (X 3 1) 2 1 = X 3 1 = X 2 = 0 which can be simplified to X = X 3 1 = X 2 = 0. Its solution set is the union of the two points of coordinates ( 1, 0, 1) and (1, 0, 1). Note that they both lie in crit(π 2, V ). When d = dim(v (f 1,..., f p )) > n p, crit(ϕ, V ) is characterized by the vanishing of all polynomials of MaxMinors(jac([f i1,..., f in d, ϕ 1,..., ϕ k ])) for all {i 1,..., i n d } {1,..., p}. Suppose that k = 1 and let L = [L 1,..., L p ] be a vector of new variables (which will stand for Lagrange multipliers) and consider the system (also known as Lagrange system) f 1 = = f p = 0, jac([f 1,..., f p ]).L = jac(ϕ 1 ). Consider the set L of its solutions in C n C p and the projection π X : (x, l) C n C p. In the case where sing(v ) =, crit(ϕ, V ) is also defined as the image of L by π X when f 1,..., f p is radical (but not necessarily equidimensional). Note that this system has a bi-homogeneous structure : all Lagrange multipliers appear with degree 1. These algebraic characterizations will allow us to compute rational parametrizations of the critical loci (or the polar varieties) when the considered critical loci are finite.

22 16 Chapitre 1. Algorithms in real algebraic geometry Now, let V be an equidimensional smooth algebraic set and ϕ : V C m be a polynomial map. The above algebraic characterizations of critical points and singular points combined with the Jacobian criterion [13, Theorem 16.19] allow us to prove that if y C k is a regular value for ϕ, then ϕ 1 (y) V is equidimensional and smooth of dimension dim(v ) k Transversality results and properties of critical points and values In this paragraph, we are interested in studying dimension properties of critical points and values. Indeed, recall that the algorithms we describe further rely on reducing the dimension of the considered problem through critical point computations. Transversality results. Transversality results are fundamental : they essentially provide informations on the dimension of the critical values of a polynomial map. The first theorem is known as Sard s theorem. Théorème (Sard s theorem). Let V C n be an equidimensional algebraic set and let ϕ : V C m be a polynomial mapping. Then ϕ(crit(ϕ, V )) is a constructible set contained in a hypersurface of C m. We continue with Thom s weak transversality theorem, specialized to the particular case of transversality to a point ; this can be rephrased in terms of critical / regular values only. Our setup is the following. Let n, d, m be positive integers and let Φ(X, Θ) : C n C d C m be a polynomial mapping. For ϑ in C d, Φ ϑ : C n C m denotes the specialized mapping x Φ(x, ϑ). Théorème (Thom s weak transversality theorem). Let W C n be a Zariski open set and suppose that 0 is a regular value of Φ on W C d. Then there exists a non-empty Zariski open subset U C d such that for all ϑ U, 0 is a regular value of Φ ϑ on W. Properties of critical points and values. The dimension of the set of critical values of polynomial mappings is controlled by Sard s Theorem [6, Chapter 9]. Ehresmann s fibration theorem (see [7, page 84] and [11] for a semi-algebraic version) provides topological informations about the fibers of proper mappings. To explain this, we need to introduce the notion of local trivial fibration. Let V C n be an equidimensional algebraic set, ϕ : V R n R k be a polynomial mapping, U be an open connected set in R k and y U. We say

23 1.2. Basic objects : algorithms and degree bounds 17 that ϕ realizes a locally trivial fibration over U if there exists a connected neighbourhood U of y and a diffeomorphism ϑ such that the following diagram commutes. φ 1 (U ϑ ) U ϕ 1 (y) U ϕ proj 2 U where proj 2 (a, b) = b. This means that geometrically, ϕ 1 (U) is perfectly described by ϕ 1 (y) U and note one can deduce from that that all fibers are diffeomorphic to each other. Théorème (Ehresmann s fibration theorem). Let V C n be an equidimensional algebraic set, ϕ : V R n R k be a polynomial mapping. Consider an open set U R k, y U and assume that ϕ is proper ; U does not contain any critical value of ϕ. Then ϕ realizes a locally trivial fibration over U. Most of the algorithms described in the next Sections rely on these results and additional properties proved in [3, 33] (see also references therein). These properties are valid in generic coordinates and are summarized below. For f Q[X 1,..., X n ] and A GL n (C), f(ax) is the polynomial obtained by applying the change of variables A to f. For simplicity, we also write f A = f(ax). If F is a finite set of polynomials in Q[X 1,..., X n ] and V = V (F) C n is the set of their common solutions, we denote by V A the set V (F A ). For 1 i n, let π i : (x 1,..., x n ) C n (x 1,..., x i ) C i. Théorème Let V C n be an equidimensional algebraic variety with sing(v ) <. There exists a non-empty Zariski open set O GL n (C) such that for all A GL n (Q) O : 1. For 1 i dim(v ), crit(π i, V A ) is either empty or it is equidimensional of dimension i For all x i C i, crit(π i, V A ) πi 1 1 (x) has dimension at most 0. dim(v )+1 3. For 1 i 2, crit(π i, V A ) is smooth at any point of V A sing(v A ). 4. For 2 i dim(v ), the restriction of π i 1 to crit(π i, V A ) is proper.

24 18 Chapitre 1. Algorithms in real algebraic geometry 5. If codim(v ) = 1, for 2 i dim(v ), crit(π 1, crit(π i, V A )) has dimension at most Generalized critical values In the context of proper mappings restricted to smooth algebraic sets, Ehresmann s fibration theorem permits to obtain important topological properties of the fibers : the considered mapping realizes a locally trivial fibration on any connected set which has an empty intersection with the set of critical values. We will see that such a fundamental property is important and useful to design algorithms for solving polynomial inequalities and for global optimization (see Subsection 1.5.1). Nevertheless, one cannot always ensure the properness property required to apply Ehresmann s theorem. In the context of non-proper mappings, this turns out to be false and we need to consider asymptotic phenomena. ( For instance, ) consider the polynomial f = X1 2 + (X 1X 2 1) 2. Since f l, l tends to 0 when l tends to and since for all x R 2, f(x) > 0, 1+l 2 0 is a global infimum of the mapping (x 1, x 2 ) R 2 f(x 1, x 2 ). This implies that it does not realize a locally trivial fibration on any interval containing 0. Nevertheless, one can check easily that 0 is not a critical value (because there is no solution of f X 1 = f X 2 at which f evaluates to 0). This shows that the properness assumption is crucial in Ehresmann s fibration theorem. We study in this paragraph the set of generalized critical values of a polynomial mapping. This object has been introduced by Kurdyka, Orro and Simon and extended in [23] and permits to generalize Ehresmann s theorem to non-proper situations. Définition [24] Let f Q[X 1,..., X n ]. A complex (resp. real) number c C (resp. c R) is an asymptotic critical value of the mapping x C n f(x) (resp. x R n f(x)) iff there exists a sequence of points (x l ) l N C n (resp. (x l ) l N R n ) such that : 1. f(x l ) tends to c when l tends to. 2. x l tends to when l tends to. 3. X i (x l ) f X j (x l ) tends to 0 when l tends to for all (i, j) {1,..., n} 2. The set of generalized critical values of a mapping is the union of the set of critical values and the set of asymptotic critical values. In the sequel, we denote by D the degree of f Q[X 1,..., X n ].

25 1.2. Basic objects : algorithms and degree bounds 19 Théorème [24] Let K(f) be the set of generalized values of x C n f(x). Then, K(f) is an algebraic set of dimension at most 0 and of cardinality dominated by D n 1. Moreover, the mapping x C n f(x) (resp. x R n f(x)) realizes a locally trivial fibration over C n \ f 1 (K(f)) (resp. R n \ f 1 (K(f) R)). f X 1 + f X 2 To illustrate the above result, consider the polynomial f = X 1 (X 1 X 2 1). Remark that f 1 (0) R 2 has 3 connected components while f 1 (e) R 2 has 2 connected components for e 0 (for e > 0, f 1 (e) is printed in red and for e < 0, f 1 (e) is printed in blue). Thus, the mapping (x 1, x 2 ) R 2 f(x) can t realize a locally trivial fibration over f 1 ((a, b)) if 0 (a, b). It is also routine to check that 0 is not a critical value of (x 1, x 2 ) R 2 f(x). Nevertheless, remark that the restriction of a generic projection, e.g. (x 1, x 2 ) x 1 x 2 to f 1 (e) for e > 0 or e < 0 has a critical point. When e varies, these critical points lie on the green line on the picture. One can observe that this critical point tends to when e 0. The same phenomenon arises when we consider the polynomial f = X1 2 + (X 1X 2 1) 2. For example, given e R, the solution set of the system f e = = 0 is crit(π, V (f e)) where π is the projection (x 1, x 2 ) x 1 x 2. When e varies these critical points lie on the red curve on the picture (the e-axis is the vertical one and the green plane is defined by E = 0). This leads us to the following observation : 0 is in the set of nonproperness of the restriction of the projection (x 1, x 2, e) e to the zero-set of f E = f X 1 + f X 2 (where E is a new variable). In [29], this phenomenon is studied and generalized to design an algorithm taking as input f Q[X 1,..., X n ] and returning a polynomial h Q[E] whose zero-set contains the set of asymptotic critical values of the mapping x C n f(x). Théorème [29] Given A GL n (Q), let C A be the Zariski closure of the solution set of f A E = f A X 2 = = f A f X n = 0, A X 1 0. Consider the projection ϕ : (x 1,..., x n, e) e. There exists a non-empty Zariski open set O GL n (C) such that for all A GL n (Q) O, the set of non-properness of the restriction of ϕ to C A contains the set of asymptotic critical values of x C n f(x). Denote by ϕ i the projection (x 1,..., x n, e) (x 1,..., x i, e). The proof of this result relies on properness properties of the restriction of ϕ i to crit(ϕ i+1, V (f A E)) for a generic A GL n (Q) (see Theorem 1.2.4).

26 20 Chapitre 1. Algorithms in real algebraic geometry FIGURE 1.2 Example f = X 1 (X 1 X 2 1) FIGURE 1.3 Example f = X (X 1X 2 1) 2 From this, one can easily design an algorithm computing the set of generalized critical values of x C n f(x). In [23], the notion of gene-

27 1.2. Basic objects : algorithms and degree bounds 21 ralized critical values is extended to polynomial mappings ϕ : V C k when V is smooth and equidimensional of dimension d and given by a generating set F of I(V ) Polynomial System Solving : the zero-dimensional case Most of the algorithms described in the next sections rely on solving zero dimensional polynomial systems. Since the goal will be to extract their real roots, it will be convenient to compute a rational parametrization of the solution set of f 1 = = f p = 0 (with f 1,..., f p in Q[X 1,..., X n ], deg(f i ) = D i and p n). Thus, finite sets of points will be represented by means of univariate polynomials, see e.g. [5, Chapter 12]. Concretely, to represent a finite algebraic set Z of C n defined over Q, we use a linear form τ = τ 1 X 1 + +τ n X n and polynomials Q = (q, q 0, q 1..., q n ) in Q[T ], with q square-free, such that Z is given by q(τ) = 0, X i = q i (τ)/q 0 (τ) (1 i n). Also, one can choose Q such that deg(q i ) < deg(q) = Z. In this case, Q will be called a zero-dimensional parametrization. It can be obtained using different methodologies (Gröbner bases, regular chains, geometric resolution) having their own advantages and drawbacks. We focus on Gröbner bases and geometric resolution algorithms. Gröbner bases and linear algebra. The most classical way to obtain such parametrizations is to use Gröbner bases [8, 14, 15]. Indeed, in the zero-dimensional case, the quotient ring Q[X 1,...,X n] f 1,...,f p is a finite dimensional vector space. The knowledge of a normal form modulo f 1,..., f p allows to reduce the computation of rational parametrizations encoding V (f 1,..., f p ) to linear algebra computations in Q[X 1,...,X n] f 1,...,f p. Gröbner bases provide such normal forms : they are a finite family of polynomials which generate f 1,..., f p with special properties (in particular, ideal membership problems can be solved with Gröbner bases). The computation of Gröbner bases can be reduced to compute row echelon forms of Macaulay matrices in several degrees (see e.g. [25, 14, 15]). Each row contains the coefficients of the monomials of polynomials obtained by multiplying the input polynomials with other monomials in order to reach some degree. The monomials are sorted using some monomial orderings (see [12]). Various monomial orderings exist ; some of

28 22 Chapitre 1. Algorithms in real algebraic geometry them, which are called elimination orderings allow to eliminate variables, i.e. to compute a basis for f 1,..., f p Q[X 1,..., X i ]. The size of the largest Macaulay matrix depends on the maximum degree we have to consider (this one is called the degree of regularity of the considered ideal). This degree is called the degree of regularity of the considered ideal and we denote it by D reg. The complexity of the Gröbner basis computation for generic inputs is ) ω ). As explained above, obtaining a rational parametrization ( (n+dreg O n from the Gröbner bases requires linear algebra operations in the finitedimensional vector space Q[X 1,...,X n] f 1,...,f p. The dimension of this vector-space is, by definition, the degree of the ideal f 1,..., f p ; it is the number of solutions counted with multiplicities. In the sequel, we will use some other features of Gröbner bases which allow us to perform geometric computations (see [17, Chapter 1]) such as computing the Zariski closure of the projection of an algebraic variety (which corresponds to computing a basis of an elimination ideal), or the Zariski closure of the set-theoretical difference of two algebraic varieties (which corresponds to computing the basis of an ideal saturated by another one). Currently, the FGB software 1 for computing Gröbner bases allows to solve systems of dimension zero with several thousands of solutions. Geometric resolutions. The geometric resolution algorithm [16, 26] (see also references therein) is a probabilistic incremental algorithm which takes advantage of evaluation properties of the input polynomial system. Thus, the input is a straight-line program evaluating the polynomial family (f 1,..., f p ). In the sequel, L denotes the length of this straight-line program. It returns rational parametrizations of zero dimensional sets obtained by intersecting each equidimensional d-dimensional component of the variety defined by the input with a generic affine subspace of dimension n d. We briefly sketch the algorithm when the input system is a reduced complete intersection (i.e. all the ideals f 1,..., f i are radical, equidimensional of co-dimension i). The algorithm computes generic points in the varieties V i defined by f 1 = = f i = 0. These generic points are obtained as the intersection of V i with n i randomly chosen hyperplanes H 1,..., H n i. Then, the jacobian matrix of (f 1,..., f i ) is evaluated at these points and a symbolic Newton/Hensel lifting is used to compute a ratio- 1. written by J.-C. Faugère

29 1.2. Basic objects : algorithms and degree bounds 23 nal parametrization of the lifted curve V i H 1 H n i 1. This curve is intersected with the hypersurface defined by f i+1 which yields a new zero dimensional set of generic points in V i+1. The process is repeated incrementally. In the case where there is no assumption on the input, it performs O(p log(d)n 4 (nl + n 4 )M(Dδ) 3 ) arithmetic operations where δ is a quantity dominated by D n (where D is the maximum degree of the input polynomials ; see [26]). When the input is a reduced complete intersection, the algorithm performs O(p(nL + n 4 )M(Dδ) 2 ) arithmetic operations (see [16]). Here δ is the maximum number of points in the intersections V i H 1 H n i. This algorithm also handles polynomial inequations Degree bounds and computation of critical points Bézout bounds. Let V C n be an algebraic set of dimension d. Its degree is the number of points obtained by intersecting V with d hyperplanes chosen generically. We denote it by deg(v ). Let W be another algebraic set. The following inequality is called Bézout s inequality : deg(v W ) deg(v ) deg(w ). When V is defined by p polynomial equations of degree D, we deduce that deg(v ) D p. Heintz proved the following variant. Defining now the strong degree of an algebraic set as the sum of the degrees of its irreducible components, and letting V = V 1 V 2 V r, the strong degree of V is dominated by Sdeg(V 1 ) max (Sdeg(V i )) i where Sdeg denotes the strong degree. Some other special bounds on degrees and number of solutions of polynomial systems exist and depend on the structure of the systems. We focus now on bounds that are well adapted to critical points. Degree bounds for critical points. Let D i = deg(f i ) and D = max(d i, i = 1,..., p). For integers (n 1, n 2,..., n k ), consider D r (n 1,..., n k ) = i 1 + +i k =r ni 1 1 n i k k.

30 24 Chapitre 1. Algorithms in real algebraic geometry Théorème If the polynomials f 1,..., f p are generic, then the degree of KKT system is bounded by D 1 D p D n p (D 1 1,..., D p 1). The above bound is sharp : it is reached when the polynomials f 1,..., f p are randomly chosen. Note also that when D = 2, these bounds are polynomial in n and singly exponential in p. Complexity of computing critical points. We leave as an exercise to the reader complexity estimates for computing a rational parametrization of a system defining critical points using their algebraic characterizations and the complexity estimates for the geometric resolution algorithm. It is more complicated to obtain complexity estimations with Gröbner bases for computing critical points. We mention the following estimate that is proved in [21]. Théorème Let f i Q[X 1,..., X n ] and D = deg(f i ) for 1 i p and consider the system defining crit(π 1, V ) below f 1 = = f p = 0, MaxMinors(jac([f 1,..., f p ], X 2 )). and its associated ideal I. If (f 1,..., f p ) is generic, the degree of regularity D reg is D(p 1) + (D 2)n + 2 ; when D = 2, computing a Gröbner basis for I requires O(n 2pω ) arithmetic operations in Q if ( D > 2 and p is ) fixed, computing a Gröbner basis for I requires O n 1 ((D 1)e) nω arithmetic operations in Q. 1.3 Computing sample points in real algebraic sets In this section, we describe some algorithms for computing sample points in each connected component of a real algebraic set. We focus on easy situations, i.e. those where the input satisfies some nice properties such as smoothness. Most of the algorithms presented here are inspired by [31]. The case of semi-algebraic sets is tackled through a combinatorial process that allows to reduce the problem to computing sample points in every connected component of several real algebraic sets. We start by describing this process following a similar strategy than the one described in [5, Chap. 13]. Next, we will consider the case of real algebraic sets which are the real trace of algebraic sets given by a reduced complete intersection. This paragraph illustrates well how we can use the computation of critical points to

31 1.3. Computing sample points in real algebraic sets 25 obtain efficient algorithms for computing sample points in real algebraic sets. The last subsection shows how the singular cases are tackled. We start by studying the case of a single equation defining a single hypersurface. This paragraph ends with an overview on how to deal with the general case and provide a more detailed view in Algorithms presented in this section allow to solve problems that involve up to 10 variables in practice Reduction to the algebraic case In this paragraph, we let S R n be the semi-algebraic set defined by f 1 = = f p = 0, g 1 0,..., g k 0 where the f i s and the g i s are polynomials in Q[X 1,..., X n ]. Théorème Let C be a connected component of S. Then there exists {i 1,..., i l } and a connected component C of the real algebraic set defined by such that C C. f 1 = = f p = g i1 = = g il = 0. Démonstration. Let C be a connected component of S and {i 1,..., i l } {1,..., k} be a maximal set (for the order induced by inclusion) such that g i1,..., g il vanish simultaneously on C. Hence, take x C such that g i1 (x) = = g il (x) = 0. Since x C S, we deduce that g j (x) > 0 for j {1,..., k} \ {i 1,..., i l }. Let C be the connected component of the real algebraic set defined by f 1 = = f p = g i1 = = g il = 0. In order to finish the proof it is sufficient to prove that for any x C, g j (x ) 0 for j {1,..., k} \ {i 1,..., i l }. We actually prove by contradiction that g j (x ) > 0 for j {1,..., k} \ {i 1,..., i l }. Indeed, for a such an integer j, assume that there exists x C and j such that g j (x ) < 0. Since C is connected, there exists a continuous function γ : [0, 1] C such that γ(0) = x and γ(1) = x (where x is the point previously defined). Since we have g j (x ) < 0, we deduce that by the intermediate value theorem that there exists t ]0, 1[ such that g j (γ(t)) = 0. This contradicts the fact that {i 1,..., i l } is a maximal set such that g i1,..., g il vanish simultaneously on C.

32 26 Chapitre 1. Algorithms in real algebraic geometry Following the above result, in order to compute sample points in each connected component of S it is sufficient to compute sample points in the real algebraic sets defined by f 1 = = f p = g i1 = = g il = 0 for all {i 1,..., i l } {1,..., k}. The case of semi-algebraic sets defined with strict inequalities is a bit more difficult. Suppose now that S R n is a semi-algebraic set defined by f 1 = = f p = 0, g 1 > 0,..., g k > 0 where the f i s and the g i s are polynomials in Q[X 1,..., X n ]. For e > 0, denote by S e the semi-algebraic set defined by f 1 = = f p = 0, g 1 > e,..., g k > e. Remark that S e S. We leave as an exercise to the reader to prove that there exists e > 0 small enough such that every connected component C of S, C S e. There are many ways to find such a real number e ; we won t describe them in this course but what is worth to remember is that, finally, computing sample points in semi-algebraic sets reduces to compute sample points in real algebraic sets The smooth equidimensional case Let (f 1,..., f p ) Q[X 1,..., X n ] generating a radical equidimensional ideal of dimension d and such that V = V (f 1,..., f p ) C n is smooth. Note that here, we do not suppose that V R n is compact ; thus considering crit(π 1, V ) is not sufficient to get sample points in each connected component of V R n in general. Geometric idea. Let C be a connected component of V R n and suppose that for all 1 i d, π i (C) is closed (for the euclidean topology). In this case, one can prove that the boundary of π i (C) is contained in the image by π i of crit(π i, C). Thus, if the boundary of π i (C) is non-empty crit(π i, C) = C crit(π i, V ) is non-empty. Note also that if π i (C) = R i, the boundary of π i (C) is empty. Thus, consider the largest integer i 0 such that π i0 (C) = R i 0 (by convention i 0 = 0 if π 1 (C) R). Then, for an arbitrary point y i0 R i 0, πi 1 0 (y i0 ) C is non-empty and its image by π i0 +1 remains closed. Hence, we deduce that πi 1 0 (y i0 ) crit(π i0 +1, C) is non-empty.

33 1.3. Computing sample points in real algebraic sets 27 y y x x FIGURE 1.4 Change of coordinates for computing sample points Finally, under the assumption that π i (C) is closed for all 1 i d and for all connected component C of V R n, one can recover at least one point in each connected component of V R n by studying πi 1 1 (y i 1) crit(π i, V ) and crit(π 1, V ) (where y i is the origin in R i ). If these algebraic sets have dimension 0, one can isolate their real roots. This process is illustrated on Figure The blue curve is the hyperbola defined by X 1 X 2 1 = 0. Using the algebraic characterization of critical points given in the previous section, it is easy to prove that there is no critical point on the restriction to the hyperbola of the projection π : (x 1, x 2 ) x 2. Remark that the projection of both connected components of the hyperbola on the X 1 -axis is open (for the Euclidean topology). Now, if a linear change of coordinate exchanges the first coordinate axis with the black lines on the two figures, the projection of these connected components becomes closed. On the figure on the left, this projection is the whole axis ; there is no real critical point and one obtains points on these connected components by taking a fiber over an arbitrary point. On the picture on the left, there is a real critical point on each connected component. Exploiting properness of projections under generic coordinates. As sketched above, it is important to ensure that for all connected component C of V R n and all 1 i d, the frontier of π i (C) is contained in π i (crit(π i, V )). This will be the case if for all 2 i d, the restriction of π i 1 to crit(π i, V ) is proper (see [31, Proposition 4]). Such properties might not hold in the initial coordinate system, but

34 28 Chapitre 1. Algorithms in real algebraic geometry hold up to a generic linear change of coordinates, by Theorem (see also [31, Theorem 1]). We will thus denote by P(A) the following assertion : for i {1,..., d + 1}, the restriction of π i 1 to crit(π i, V A ) is proper. Under this condition, the following theorem enables to compute one point on each connected component of V R n. Théorème [31] Let A GL n (C) be such that P(A) holds. Let y d = (y 1,..., y d ) be any point in R d. For 1 i d 1, define y i = (y 1,..., y i ) R i. For i = 0, we formally define π0 1 (y 0) as C n. Then, the algebraic sets crit(π i, V A ) π 1 i 1 (y i 1) for i {1,..., d + 1}, are either empty or zero-dimensional. Their union meets every connected component of V A R n. Hence, the algorithm consists in finding A satisfying P(A) and solving the systems defining crit(π i, V A ) π 1 i 1 (y i 1), for i {1,..., d + 1}. Degree bounds. Let D i = deg(f i ) for 1 i p and D = max(deg(f i ), 1 i p). Consider the projection ϕ i : (x 1,..., x n ) x i. Remark now that crit(π i, V A ) π 1 i 1 (y i 1) = crit(ϕ i, V A ) π 1 i 1 (y i 1). We use this equality to give degree bounds on the sections of polar varieties which appear in Theorem Suppose first that d = n p. In this case, one can apply Theorem Recall that for integers (n 1, n 2,..., n k ), D r (n 1,..., n k ) denotes i 1 + +i k =r ni 1 1 n i k k. Then the degree of crit(ϕ i, V A π i 1 (y A )) is bounded by D i = D 1 D p D n i 1 p (D 1 1,..., D p 1). Recall that D i D 1 D p (D 1) n i p( n i 1 n i p). Thus, one can deduce that in the quadratic case, the output of this algorithm is polynomial in the number of the variables. This shows that, at least in terms of size of the output, the algorithm behaves well since it matches the best complexity bounds in this case [18]. If d < n p, one can only apply special Bézout bounds (that are called bi-homogeneous Bézout bounds, see [34]) and we get that the degree of crit(ϕ i, V A π i 1 (y A )) is bounded by D i = D 1 D p (D 1) n p ( n 1 n p ).

35 1.3. Computing sample points in real algebraic sets 29 Once again, we obtain in the quadratic case degree bounds which are polynomial in the number of variables. The complexity of the algorithm is the sum of the complexity of solving d + 1 zero-dimensional polynomial systems of respective degrees bounded by D i (or D i in the generic case). Supposing that this is done in polynomial time in the degree we get D O(1) D O(1) n p+1. By applying Theorem to the computation of critical points of the square of distance functions, we would get a degree bound equal to D D n p+1, and thus a complexity (D D n p+1 ) O(1). Thus, exploiting properness properties of projections permits to split computations of sample points in smooth equidimensional situations. Complexity analysis. Using the geometric resolution algorithm to solve the zero dimensional systems built by the above algorithm, one can analyze its complexity. The geometric resolution algorithm is incremental. So we will consider the polynomials f1 A,..., f p A first and then add the minors of jac([f 1,..., f p ]) required to define crit(π d, V A ), and then crit(π d 1, V A ) and so on until crit(π 1, V A ). Let G be the sequence of these polynomials ordered as described and S 1 = ( )( p n n d n d) the total number of minors that we need to consider. Below, we denote by δ the maximum of the algebraic degrees (see [26]) of the ideals defined incrementally by a given prefix subsequence If f 1,..., f p are of degree bounded by D, then δ is bounded by n(d(n d)) n [26, page 4] (note that here one can not apply the degree bounds on the critical loci because we consider the polynomials incrementally). We can now state the complexity result. Théorème [31, Theorem 3] Let f 1,..., f p be polynomials of degree bounded by D in Q[X 1,..., X n ], given by a Straight-Line Program of length L. Suppose that f 1,..., f p is a radical, equidimensional ideal and that V = V (f 1,..., f p ) C n is smooth of dimension d. There exists a probabilistic algorithm computing a family of geometric resolutions, the reunion of whose real zeros contains at least one point in each connected component of V R n. In case of success, its complexity is within O ((p + S 1 ) log(d)n 5 (S 1 L(n d) 4 + n 3 (n d) 4 )M (D(n d)δ) 3) arithmetic operations.

36 30 Chapitre 1. Algorithms in real algebraic geometry The probabilistic aspects come from putting the system in general position, and also appear during the execution of the algorithm of [26]. From Theorem 1.2.4, the probability of success depends on the choice of points inside non-empty Zariski open sets Deformation techniques and singular algebraic sets We investigate now algorithms for computing sample points in the real counterpart of singular hypersurfaces. Deformation techniques have been introduced to deal with singular situations within a good theoretical complexity. The idea is rather simple. Starting from a singular situation, one can regularize it by introducing a formal parameter that is an infinitesimal and make the set smooth ; next perform sample point computations on this smooth set and finally let the parameter tend to 0 to recover sample points on the initial set under consideration. Before entering into more details, we need to introduce this notion of infinitesimals. Preliminaries on infinitesimals. An infinitesimal ε over R is a transcendental element such that for any 0 < ε < x for any positive element x R. Let K be a field containing Q (e.g. R or C). Let ε be an infinitesimal and let K ε stand for the Puiseux series field. We say that z = i i 0 a i ε i/q K ε is bounded over K if and only if i 0 0. We say that z = (z 1,..., z n ) K ε n is bounded over K if each z i is bounded over K. Given a bounded element z K ε, we denote by lim ε 0 z the number a 0 K. Given a bounded element z K ε n, we denote by lim ε 0 z the point (lim ε (z 1 ),..., lim ε (z n )) K n. Given a subset A K ε n, we denote by lim ε (A) the set {lim ε (z) z A and z is bounded}. Given a semi-algebraic (resp. constructible) set A R n (resp. A C n ) defined by a quantifier-free formula Φ with polynomials in R[x 1,..., x n ], we denote by ext(a, R ε ) (resp. ext(a, C ε )) the set of solutions of Φ in R ε n (resp. C ε ). We refer to [5, Chapter 2.6] for precise statements of these notions. We can clarify now the deformation process that has been sketched in the beginning of this subsection. Let f be a polynomial in Q[X 1,..., X n ] of degree D ; we denote by V C n the algebraic set defined by f = 0.. Then, for e R ε, we denote by V ε C ε n the algebraic set defined by f ε = 0. Lemme The algebraic set V ε is smooth.

37 1.3. Computing sample points in real algebraic sets 31 Démonstration. Consider the map x C n f(x). By Sard s theorem, the set of critical values of this map is finite and hence algebraic over Q (since f has coefficients in Q). Since ε is transcendental, it does not belong to the set of critical values of this map. This implies that at any point x such that f(x) = ε, the gradient vector of f is not 0. Our conclusion follows. Geometric idea and overview of the algorithm. Following the discussion on the degree bounds in Subsection 1.3.2, one may want to reduce the computation of sample points in V R n to the computation of limits of sections of the polar varieties crit(π i, V ε ). As in Subsection 1.3.2, this is possible under some properness properties of these polar varieties which can be satisfied under a generic linear change of variables (see [28]). To do that, one could consider to tackle the infinitesimal ε as a formal parameter and perform the computations over the field Q(ε) before taking the limit at 0. However, this makes the computations heavy in practice since the arithmetic of the ground field Q(ε) becomes now much more expensive than the one of Q. To solve this issue, we avoid to perform computations over Q(ε) for computing lim ε crit(π 1, Vε A ) (with A GL n (Q)). Let L be a new variable and consider the system f A X 2 = = f A X n = 0, L f A X 1 = 1. Let L(π 1, V A ) be its solution set, π X be the projection on the X-space, and C(π 1, f A ) be the Zariski closure of π X (L(π 1, V A )). It is straightforward to figure out that any point lying in lim ε crit(π 1, Vε A ) lies in C(π 1, f A ) V : the points in lim ε crit(π 1, Vε A ) which satisfy f A X 1 = 0 lie in the set of non-properness of the restriction to π 1 to L(π 1, V A ), the others lie in π X (L(π 1, V A )). Note also that one can bound the degree of C(π 1, V A ) by (D 1) n 1 and that an algebraic representation of it can be obtained by means of Gröbner bases computations (using monomial block-orderings) or computations of geometric resolutions (see [16] and [35]). From this, one can obtain algebraic representations of C(π 1, V A ) V. We describe now the algebraic objects encoding sections of the considered polar varieties. Given A GL n (Q) and 1 i n 2, Ii A denotes the ideal generated by X 1,..., X i, L f A f A 1,,..., f A X i+1 X i+2 X n and J A i denotes the ideal I A i Q[X 1,..., X n ]. If i = 0, I A 0 denotes the ideal L f A X 1 1, f A X 2,..., f A X n

38 32 Chapitre 1. Algorithms in real algebraic geometry and J A 0 denotes I A i Q[X 1,..., X n ]. If i = n 1, we set J A 0 = X 1,..., X n 1, f A. Théorème [28] There exists a non-empty Zariski open set O GL n (C) such that for all A GL n (Q) O the following holds : for 0 i n 2, Ii A has dimension at most 1 ; for 0 i n 1, Ji A has dimension at most 0 ; for 0 i n 1, the algebraic variety associated to Ji A contains crit(π i, Vε A ) πi 1 1 (0) ; the union of the algebraic varieties associated to J0 A,..., J n 1 A meets every connected component of V R n. As in Subsection 1.3.2, a matrix A will lie in the non-empty Zariski open set O if a properness property (that can be checked with NonProperness) is satisfied. The algorithm which relies on the above result computes rational parametrizations of J0 A,..., J n 1 A. This can be done using either Gröbner bases or geometric resolutions. Degree bounds. We show now why the degree of J0 A is always less than the Bézout bound D(D 1) n 1 when dim(sing(v A )) > 0. Let d be the sum of the degrees of the equidimensional components of sing(v A ) of positive dimension. Remark that the degree of the curve C defined as the Zariski closure of the solution set of : f A X 2 = = f A X n = 0, f A X 1 0 is bounded by (D 1) n 1 d. Thus, the degree of C V A is bounded by D ( (D 1) n 1 d ) which is always less than D(D 1) n 1 when d > 0. Taking into account the above discussion and performing a careful analysis of degree bounds for the objects introduced in Theorem leads to the following result. Théorème [28] Let H 1,..., H n 2 be generic hyperplanes of Q n. The number of connected components of V R n is bounded by D(1 + (D 1) + + (D 1) n 1 (d 0 + +d n 2 )), where d i (resp. d 0 ) denotes the sum of the degrees of the positive-dimensional components of the singular locus of V ( i j=1 H i) (resp. V ). Complexity issues. The following complexity result is obtained by using geometric resolution algorithms and results in [35] to compute rational parametrizations of the algebraic varieties associated to Ii A and Ji A for 0 i n 1.

39 1.4. Connectivity in real algebraic sets 33 Théorème [28] Let f be a polynomial in Q[X 1,..., X n ] of degree D, encoded by a straight-line program of length L and V C n be the hypersurface defined by f = 0. There exists a probabilistic algorithm which computes at least one point in each connected component of V 0 R n within O ( n 3 (L + n 2 ) M(D.δ) 3) arithmetic operations in Q where δ is the maximal degree of the intermediate algebraic varieties studied during the incremental process and is bounded by D.(D 1) n 1. Generalization. We explain now how the above strategy can be extended to general polynomial systems. Let V C n be the algebraic set defined by f 1 = = f p = 0 with f i Q[X 1,..., X n ] for 1 i p. For positive reals a 1,..., a p, consider the semi-algebraic set S ε R ε defined by : a 1 ε f 1 a 1 ε,..., a p ε f p a p ε. It is clear that ext(v R n, R ε ) is contained in S ε and that V R n = lim ε S ε. This latter inequality can be made more precise : for any connected component C of V R n, there exist connected components C 1,..., C r of S ε such that C = lim ε (C 1 C r ). Hence, one can compute sample points in each connected component of S ε and take their limit to obtain sample points in V R n. To do that, one uses the reduction to algebraic sets explained in the first paragraph of this section. Using Sard s theorem in a similar way to the proof of Lemma 1.3.4, one can prove that for a generic choice of (a 1,..., a p ) in Q p the algebraic sets defined by f i1 a i1 ε = = f il a il ε = 0 are smooth (and the ideal generated by these equations is radical and equidimensional). Computing critical points on these sets can be made by using similar techniques to those we just described to avoid to manipulate ε as a formal parameter. 1.4 Connectivity in real algebraic sets The algorithms for computing sample points in real algebraic sets (or semi-algebraic sets) may return a large number of points (e.g ). It is often useful to reduce the size of the output by returning a unique point per connected component of the considered semi-algebraic set and one

40 34 Chapitre 1. Algorithms in real algebraic geometry way to do that is to answer connectivity queries. This section focuses on computing roadmaps which allow to answer connectivity queries. We focus on roadmap computations for sets of the form V R n, where V C n is a smooth algebraic set and V R n is compact. These sets are defined by a system F = (f 1,..., f p ) in R[X 1,..., X n ], with p < n, which satisfies assumption (A) : (a) the ideal f 1,..., f p is radical ; (b) V = V (f 1,..., f p ) is equidimensional of positive dimension d = n p > 0 ; (c) sing(v ) is finite ; (d) V R n is bounded. In Subsection 1.4.1, we recall the original definition of roadmaps and explain some slight modifications that have been brought to this notion in [33]. Subsection describes a variant of Canny s algorithm running in (nd) O(n(n p)) given in [27]. This variant of Canny s algorithm has practical performances that reflect the complexity breakthrough obtained by Canny. This has been done by substituting computations in algebraic extension fields arising in Canny s original algorithms by computations where some variables are instantiated to rational numbers (and the number of these instantiations is well-controlled). Subsection introduces a new connectivity result that gives more freedom for the design of roadmap algorithms. This result appears first in [33] ; it has been used to obtain an algorithm running in time (nd) O(n1.5 ) but is also a key step towards algorithms running in time (nd) O(n log 2 (n p)) (see [34] and references therein). For both families of algorithms, the strategy is similar. The algorithms start by computing a critical set (i.e. a polar variety) which, under our assumptions, will have a non-empty intersection with all connected components of V R n. However, these intersections may not be connected and these connectivity failures will be repaired by slicing V R n with linear spaces of a suitable dimension Definitions, data-representation and use of the output The original definition (found in [5]) is as follows. Let S R n be a semi-algebraic set. A roadmap for S (in the sense of [5]) is a semi-algebraic set R of dimension at most 1 which satisfies the following conditions : RM 1 R is contained in S. RM 2 Each connected component of S has a non-empty and connected intersection with R.

41 1.4. Connectivity in real algebraic sets 35 RM 3 For x R, each connected component of S x intersects R, where S x is the set of points of the form (x, x 2,..., x n ) in S. We modify this definition (in particular by discarding RM 3 ), for the following reasons. First, it is coordinate-dependent : if R is a roadmap of S, it is not necessarily true that R A is a roadmap of S A, for A GL n (Q). Besides, one interest of RM 3 is to make it possible to connect two points in S by adding additional curves to R : condition RM 3 is well-adjusted to the procedure given in [5], which we do not use here. Hence, we modify in the definition of roadmaps. Recall that the algorithms described below do not deal with semi-algebraic sets, but only with real counterparts of algebraic sets V C n. The definition we give here, like the previous one, allows us to count connected components and to construct paths between points in V R n. Also, we generalize the definition to higher-dimensional roadmaps, since our algorithm computes such objects. Thus, we say that an algebraic set R C n is a roadmap of V if : RM 1 Each semi-algebraically connected component of V R n has a non-empty and semi-algebraically connected intersection with R R n. RM 2 The set R is contained in V. Next, we say that R is an i-roadmap of V if in addition we have : RM 3 The set R is either i-equidimensional or empty. Finally, it will be useful to add a finite set of control points P to our input, e.g. to test if the points of P can be connected on V R n. Then, R is a roadmap (resp. i-roadmap) of (V, P) if we also have : RM 4 The set R contains P V R n. Data representation. The output of our algorithms is a rational parametrization of an algebraic curve. If Z C e is an algebraic curve defined over Q, a one-dimensional parametrization of Z consists in polynomials Q = (q, q 0,..., q e ) in Q[U, T ] and two linear forms τ = τ 1 X τ e X e and η = η 1 X η e X e with coefficients in Q, with q square-free and monic in U and T, gcd(q, q 0 ) = 1, and such that Z is the Zariski closure of the set defined by q(η, τ) = 0, X i = q i(η, τ) q 0 (η, τ) (1 i e), q 0(η, τ) 0. Given a parametrization Q, the corresponding curve Z is denoted by Z(Q). The degree of Z is written δ Q ; then, all polynomials in Q can, and will, be taken of degree δ O(1) Q (see [35]). We will also request that the degree of q is δ Q both in U and T.

42 36 Chapitre 1. Algorithms in real algebraic geometry In the zero dimensional case, sets of points Z C e are represented in a standard way by a rational parametrization Q ; we will denote by δ Q the cardinality of Z which bounds the degrees of the polynomials appearing in Q. Again, Z(Q) will denote the set Z defined by Q. In both zero- and one-dimensional cases, if Q represents a set of points Y in C e, with variables X 1,..., X e, it will be helpful to write Q(X 1,..., X e ) to indicate what variables are used ; Q is defined over Q if all polynomials in it have coefficients in Q. Finally, a parametrization of the empty set consists by convention of the unique polynomial Q = ( 1). Using the output. Let us briefly sketch how to use the output of the algorithms described below to answer connectivity queries for points in an algebraic set V. Given a set of control points P of cardinality 2, the onedimensional parametrization Q = (q, q 0,..., q n ) only describes an open dense subset of a roadmap containing P. It is possible to recover the finitely many missing points, by means of a zero-dimensional parametrization Q, using Puiseux expansions at the points where both q and q 0 vanish. Since all polynomials in Q have degree δ O(1) Q, this can be done in time δ O(1) Q. Given this, one can compute a Cylindrical Algebraic Decomposition adapted to the constructible sets defined by Q and Q. In view of the simple shape of the defining polynomials, this takes time δ O(1) again. To compute adjacencies between cells, we use the algorithm of [36], which takes time δ O(1) Q using again some Puiseux expansion algorithm Practical improvement of Canny s algorithm We consider now a system F = (f 1,..., f p ) in R[X 1,..., X n ], with p < n satisfying assumption (A) and supposing additionally that sing(v ) =. Recall that, under (A), d = n p is the dimension of V (F) and crit(π i, V ) is defined by F, MaxMinors(jac(F, [X i+1,..., X n ])) (see Subsection 1.2.2). In the sequel, denotes the set of minors MaxMinors(jac(F, [X i+1,..., X n ])). We say that F satisfies condition (B) 2 if the following holds : (a) for any x C d, V π 1 d (x) is finite ; (b) either W 2 = crit(π 2, V ) is empty, or W 2 is equidimensional, of dimension one, and for any x C, W 2 π1 1 (x) is finite ; (c) crit(π 1, V ) and crit(π 1, W 2 ) are finite ; (d) for x in W 2 sing(v ), jac x ([F, ], [X 1,..., X n ]) has rank n (i 1). By Theorem 1.2.4, these properties can be ensured by a generic change of variables. We define now Q

43 1.4. Connectivity in real algebraic sets 37 FIGURE 1.5 Roadmaps using Canny s algorithm C = crit(π 1, V ) crit(π 1, W 2 ), which is finite under (A) and (B) 2 ; Q = π 1 (C) and V Q = V π1 1 (Q). Canny s algorithm relies on a connectivity result that can be restated as follows. Théorème Under assumptions (A) and (B) 2, the following holds : 1. V Q W 2 is a roadmap of (V, P) ; 2. V Q is either empty or (d 1)-equidimensional ; 3. V Q W 2 = W 2 π1 1 (Q) is finite ; 4. for all x C, the system (f 1,..., f p, X 1 x 1 ) satisfies assumption (A) else it has dimension at most 0. Note that Q is a finite set of points encoded by a univariate polynomial Q Q[X 1 ]. Canny s algorithm [9] and its improvements (see [5] and references therein) apply recursively the above result and will perform computations modulo Q. Hence, the degree of Q plays an important role in the practical behavior of the algorithm. Since we are only interested in real roots of Q, a natural idea, illustrated by the figure aside, consists in isolating the real roots ρ 1 < < ρ l of

44 38 Chapitre 1. Algorithms in real algebraic geometry Q, pick up rational numbers between these real roots ρ 1 < r 1 < ρ 2 < r 2 < < r l 1 < ρ l and perform the recursion with X 1 = r i for 1 i l 1. Given a finite set Q = {ρ 1,..., ρ l } R with ρ 1 < < ρ l, Sep(Q) denotes a set of rational numbers r 1,..., r l 1 such that ρ 1 < r 1 < ρ 2 < r 2 < < r l 1 < ρ l. The strategy described above is valid because Canny s connectivity result can be extended as follows : Théorème [27] Under assumptions (A) and (B) 2, the following holds : 1. V Sep(Q) W 2 is a roadmap of (V, P) ; 2. V Sep(Q) is either empty or (d 1)-equidimensional ; 3. V Sep(Q) W 2 = W 2 π1 1 (Sep(Q)) is finite ; 4. for all r Sep(Q), the system (f 1,..., f p, X 1 r) satisfies assumption (A) and V (f 1,..., f p, X 1 r) has an empty singular locus. The idea of the proof is the following. Let x π 1 (crit(π 1, W 2 )) and a > 0 be small enough such that (x a, x + a) contains no point of π 1 (crit(π 1, W 2 )). It is proved in [5] that there exists a one-to-one correspondence between the connected components of (V R n )π1 1 (x) and those of (V R n )π1 1 ((x a, x + a)). Hence the connectivity failures that are repaired by taking fibers slices of V above x can be repaired by taking slices above x a and x + a. Finally, the idea of the algorithm is to compute first W 2 and Sep(Q). Then, one applies recursively Theorem on V Sep(Q). To do that, we need to ensure that V Sep(Q) satisfies (A) and (B) 2. By Theorem a generically chosen linear change on variables ensures this latter property. [27, Proposition 1.5] shows that V Sep(Q) satisfies (B) 2 if the initial system F is in generic coordinates and this will be the case for all algebraic sets encoded by polynomial families appearing in the recursive calls. Overview of the algorithm, practical performances and complexity. The algorithm takes as input a system F Q[X 1,..., X n ] satisfying (A). It first chooses randomly a linear change of variables A GL n (Q). Then, it performs a call to an internal recursive routine. This routine, which is called SubRoadmap in the sequel, takes as input F A and will output a one dimensional parametrization R. Then, the algorithm outputs R A 1. The subroutine SubRoadmap works as follows. It first computes a rational parametrization R 1 of W2 A = crit(π 2, V A ) and a polynomial Q Q[X 1 ] such that ( V (Q) = π 1 crit(π1, V A ) crit(π 1, W2 A ) ).

45 1.4. Connectivity in real algebraic sets 39 Then, it computes Sep(V (Q)). Recursive calls to SubRoadmap are performed on the polynomial families Substitute(X 1 = r i, F A ) for 1 i l 1 and return rational parametrizations R i to which one adds the equation X 1 = r i. Denoting by R 2 a rational parametrization encoding Z(R 1 ) Z(R l 1 ), SubRoadmap returns a rational parametrization encoding Z(R 1) Z(R 2 ). In the worst case, the number l above is O(nD) n, which is the degree of the polynomial Q. Here, the number of recursive calls that is performed depends on the number of real roots of Q. Note also that at each recursive call, the dimension decreases by 1, thus the depth of the recursion is d = dim(v (F)). Finally, using the complexity results of [16] (see Subsection 1.2.5), the following complexity estimates is given in [27]. Théorème [27] Consider a family F = (f 1,..., f p ) Q[X 1,..., X n ] satisfying (A). Suppose that sing(v (F)) =. One can compute a 1-roadmap of V (F) of degree O((nD) n(n p) ) in probabilistic time O(n n+9 (nd) n M((nD) n(n p) ) 3 ). The algorithm described above has been implemented. Our implementation uses FGb for Gröbner bases computations and RS for isolating real roots of zero dimensional systems. A large number of variables can be handled by this preliminary implementation. The computing time depends of course on the depth of the recursion and the maximum number of real roots of the polynomials Q arising in the recursive calls. Consider h as the the maximum of these numbers of real roots : in the worst case h O((nD) n ) but in practice h is most of the time much smaller. The degree of the computed roadmap is O(h n p (nd) n ). Note that our implementation tackles systems that are out of reach of the best implementations of Cylindrical Algebraic Decomposition ; it seems also that its practical behavior depend heavily on h. These systems have a moderate size, but recall that our implementation is a preliminary one. This shows that one can expect in the future practically efficient implementations of roadmap algorithms A baby steps/giant steps approach for smooth bounded hypersurfaces To compute a roadmap in a d-dimensional algebraic set V, all algorithms based on Canny s approach start by computing a critical curve and repair connectivity failures of this critical curve by considering additio-

46 40 Chapitre 1. Algorithms in real algebraic geometry FIGURE 1.6 Example of divide-and-conquer strategy to compute roadmaps nally (d 1)-dimensional varieties contained in V. Thus, the depth of the recursion is d, which leads to the complexity described above. In order to improve the complexity, a simple idea is to start by computing i-dimensional critical sets (with i > 1) and repair connectivity failures by adding d i-dimensional algebraic sets contained in V. A suitable choice of i may reduce the depth of the recursion and hence improve the complexity of computing roadmaps. To illustrate it, let us consider a smooth hypersurface V in C 4 defined by f = 0 with f Q[X 1, X 2, X 3, X 4 ]. Suppose that the real counterpart of V is bounded. On Figure 1.6, π 3 (V ) is the set of points in the closure of the interior of the red ball not in the interior of the green ball. These balls are the images by π 3 of W 3 = crit(π 3, V ). Suppose that the green line on Figure 1.6 is the X 1 -axis and let Q ( = π 2 (crit(π 1, V ) crit(π 1, W 3 )) R 4. Observe now that crit(π 3, V ) V π 1 2 (Q)) is connected (unhappily on the picture one can only observe that the projection of this latter set is connected). Note now that R 2 = ( V π2 1 (Q)) (in blue on the Figure) has already dimension 1, while crit(π 3, V ) has dimension 2. Thus, one can compute a 1-dimensional roadmap R 1 of crit(π 3, V ) with one recursive call. Note also that we have to add some linking curves in orange on the Figure to ensure that R 1 R 2 is a roadmap of V. Finally, one can see that by considering intermediate higher-dimensional polar varieties to get higher-dimensional roadmaps, the depth of our recursion is only 1. This is better than Canny s approach for which the depth of the recursion is 2 on this example. This illustrates an other way of exploiting topological properties of polar varieties and critical points to speed up the computations. We develop below this idea of considering intermediate higherdimensional polar varieties to get higher-dimensional roadmaps. We

47 1.4. Connectivity in real algebraic sets 41 still consider a system F = (f 1,..., f p ) in R[X 1,..., X n ], with p < n satisfying assumption (A). Recall that, under (A), d = n p is the dimension of V (F) and W i = crit(π i, V ) is defined by F, MaxMinors(jac(F, [X i+1,..., X n ])) (see Subsection 1.2.2). In the sequel, denotes the set MaxMinors(jac(F, [X i+1,..., X n ])). Assume d 2 and fix i in {2,..., d}. Then, we consider a generalization of the condition (B) 2 described above. We say that F satisfies condition (B) i if the following holds : (a) for any x C d, V π 1 d (x) is finite ; (b) either W i is empty, or W i is (i 1)-equidimensional and for any x C i, W i πi 1 1 (x) is finite ; (c) crit(π 1, V ) and crit(π 1, W i ) are finite ; (d) for x in W i sing(v ), jac x ([f, ], [X 1,..., X n ]) has rank n (i 1). By Theorem 1.2.4, these properties can be ensured up to a generic change of variables for some values of p and i (but not all). Finally, we consider a finite subset of points P in V ; with this convention, we define C = crit(π 1, V ) crit(π 1, W i ) P, which is finite under (A) and (B) i ; Q = π i 1 (C) and V Q = V πi 1 1 (Q). Théorème [33] Under assumptions (A) and (B) i, the following holds : 1. V Q W i is a roadmap of (V, P) ; 2. V Q is either empty or (d i + 1)-equidimensional ; 3. V Q W i = W i πi 1 1 (Q) is finite ; 4. for all x C i 1, the system (f 1,..., f p, X 1 x 1,..., X i 1 x i 1 ) satisfies assumption (A). Theorem is exactly Theorem with i = 2. The freedom on the choice of i will be used to reduce the depth of the recursion. As in the previous paragraph, the idea of the algorithm in [33] will roughly be to compute the sets W i and V Q, update the control points by computing V Q W i and to recursively compute a roadmap R 1 of (V Q, P (V Q W i )) on the one hand and a roadmap R 2 of (W i, P (V Q W i )) on the other hand. Thanks to the proposition below, R 1 R 2 is a roadmap of (V, P). Proposition [33] Let R 1 and R 2 be algebraic sets such that R 1 R 2 is a roadmap of (V, P), and such that R 1 R 2 is finite. Let R 1 and R 2 be 1-roadmaps of respectively (R 1, (R 1 R 2 ) P) and (R 2, (R 1 R 2 ) P). Then R 1 R 2 is a 1-roadmap of (V, P).

48 42 Chapitre 1. Algorithms in real algebraic geometry Overview of the algorithm. The core routine MainRoadmap (see [33, Section 5.5]) takes as input f Q[X 1,..., X n ], a zero dimensional parametrization B(X 1,..., X e ) defining a finite set B C e and a zero dimensional parametrization P (X 1,..., X n ) defining control points P. The couple (f, B) defines an algebraic variety V B = V (f) πe 1 (B) of dimension d = n e 1 that satisfies (A) and the routine is expected to return a roadmap of (V B, P). By Theorem 1.2.4, one may suppose that V B satisfies (B) d, up to a generic enough linear change of variables leaving invariant the (X 1,..., X e )- coordinates. Consequently, one can apply Theorem with i = d taking care of the fact that the (X 1,..., X e )-projection of V B is a zero dimensional set. Thus, we will consider the projections π e,i : (x 1,..., x n ) (x e+1,..., x i ), π e,i 1 : (x 1,..., x n ) (x e+1,..., x i 1 ) and π e,e+1 : (x 1,..., x n ) x e+1 in place of π i, π i 1 and π 1. Following Theorem 1.4.4, this leads to consider a polar variety W i,b in V B, a new finite set of points Q in C e+i 1 which enable to define V Q and a new set of control points P = P (V Q W i,b ) in order to be able to apply Proposition The degree of all these objects increases by a factor (δ B + δ P )(nd) O(n) (see [33]). Since dim(w i,q ) d, a variant of Canny s algorithm (see [33, Section 5.4]) returns a roadmap of (W i,q, P ) of degree at most (δ B + δ P )(nd) O(n d) in probabilistic time (δ B +δ P ) O(1) (nd) O(n d). Also, a call to MainRoadmap with input f, and parametrizations defining Q and P will return a roadmap of (V Q, P ) of degree bounded by (δ B + δ P )(nd) O(n d) in probabilistic time (δ B + δ P ) O(1) (nd) O(n d). By the definition of P and Proposition 1.4.5, R 1 R 2 is a roadmap of (V Q, P). It is straightforward to figure out that the depth of the recursion is n, which yields the following complexity result. Théorème [33] Given f square-free in Q[X 1,..., X n ] such that V (f) has a finite number of singular points and V (f) R n is bounded, and given a set P of cardinality δ P, one can compute a 1-roadmap of (V (f), P) of degree δ P (nd) O(n1.5) in probabilistic time δ P O(1) (nd) O(n1.5). 1.5 Towards quantifier elimination Algorithms computing sample points in algebraic sets or in fulldimensional semi-algebraic sets described in Section 1.3 rely on computations of critical points. In some sense, we solve global optimization problems to grab points in semi-algebraic sets. Thus, it is natural to investi-

49 1.5. Towards quantifier elimination 43 gate the way one can use the material described previously to solve global polynomial optimization problems. In symbolic computation, polynomial optimization problems are solved via quantifier elimination (see [5, Section 14.4]). The main difficulty is to tackle the cases where the global infimum is not reached. In this context, my goal has been to provide new quantifier elimination algorithms taking advantage of the structural properties arising in these applications. Subsection deals with polynomial optimization [30] using the notion of generalized critical values (see Subsection 1.2.4) : the topological properties of generalized critical values (see Theorem 1.2.6) are used to obtain an algorithm solving global optimization problems. Subsection describes an algorithm for quantifier elimination under properness and regularity assumptions which arise in a wide range of applications. It allows to solve application that are out of reach of the best implementations of Cylindrical Algebraic Decomposition. This section summarizes results from [20] Polynomial Optimization Let f Q[X 1,..., X n ] and consider the following global optimization problem inf x R n f(x). We describe an approach which relies on the critical point method and quantifier elimination [5, Chapter 14] for solving this problem. It allows us to decide if a given polynomial f Q[X 1,..., X n ] has a global infimum over the reals and if this is the case, it provides a polynomial Q Q[T ] and an interval (a, b) such that (a, b) contains a real root which is the global infimum of f. Note that this encoding allows to isolate the global infimum with an arbitrary precision. In this framework, most of the problems come from the fact that multivariate polynomials may not reach their extrema, as is the case for the map (x 1, x 2 ) x (x 1x 2 1) 2 whose global infimum is 0 and is not reached at a critical point (see Subsection 1.2.4). To deal with these asymptotic phenomena, one can relate the set of generalized critical values (see Definition 1.2.5) to the global infimum. Théorème [30] Let f Q[X 1,..., X n ] and E = {e 1,..., e l } be the set of real generalized critical values of the mapping x R n f(x) such that e 1 <... < e l. Consider {r 0,..., r l } a set of rationals such that r 0 < e 1 < r 1 <... < e l < r l. The infimum of f over R n is finite if and only if there exists i 0 {1,..., l} such that for all j < i 0 the real algebraic set {x R n f(x) = r j } is empty

50 44 Chapitre 1. Algorithms in real algebraic geometry the real algebraic set {x R n f(x) = r i0 } is not empty. In this case inf x R n f(x) = e i0. The above result is proved by exploiting topological properties of generalized critical values (see Theorem 1.2.6). Indeed, suppose that the infimum of f over R n is finite ; we denote it by f. Then, by definition, there exists ε R small enough such that f 1 (e) R n = for all e (f ε, f ) and f 1 (e) R n for all e (f, f + ε). This implies that f does not realize a locally trivial fibration over an interval containing f, which implies that f is a generalized critical value of x R n f(x). Moreover note that for all 0 i l, the algebraic sets f 1 (r i ) are smooth hypersurfaces (since the rational number r i can t be a critical value of the mapping x f(x)). Since we already have algorithms for computing generalized critical values (see Subsection 1.2.4) and for deciding the emptiness of smooth algebraic sets (see Subsection 1.3.2), one can obtain an algorithm for solving global optimization problems. It starts by computing a univariate polynomial Q whose real roots contains the set of generalized critical values of x f(x), picks up a rational number r between each real root of Q and decides the emptiness of the algebraic sets defined by f r = 0 (note that such algebraic sets are smooth since r is not in the set of critical values of f). Using results in Sections and 1.2.5, one can estimate the running time of a probabilistic version of this algorithm. Théorème [30] Let f Q[X 1,..., X n ] of degree D and L be the length of a straight-line program evaluating f. There exists an algorithm which decides if inf x R n f(x) is finite and, in case of finiteness, computes a non-zero univariate polynomial Q whose set of real roots contain it in probabilistic time O(D n n 3 (L + n 2 )M(D 2 (D 1) (n 1) ) 2 ) Variant real quantifier elimination Let X = [X 1,..., X n ], Y = [Y 1,..., Y r ], and let Π Y be the canonical projection (x, y) y. Consider F = (f 1,..., f p ) and G = (g 1,..., g s ) in Q[X, Y], S R n+r be the semi-algebraic set defined by f 1 = = f p = 0, g 1 > 0,..., g s > 0 and the quantified formula Ψ X R n f 1 (X, Y) = = f p (X, Y) = 0, g 1 (X, Y) > 0,..., g s (X, Y) > 0. Classical quantifier elimination algorithms return a quantifier-free formula describing Π Y (S). In [20], we consider a variant of the classical quantifier elimination problem. We strengthen the requirements on the input by assuming that :

51 1.5. Towards quantifier elimination 45 The ideal f 1,..., f p is radical and equidimensional of codimension c and the complex variety V (f 1,..., f p ) C n+r is smooth. The restriction of Π Y to the real variety of V (f 1,..., f p ) R n+r is proper. Note that here we do not suppose that the intersection of V (F) with V (g i ) (for 1 i s) is transverse and smooth. In particular, this implies that the intersections of V (F) with V (g i ) may be singular. We weaken the output by returning a quantifier-free formula Φ (defining a semi-algebraic set S R r ) which is almost equivalent to Ψ, that is, S Π Y (S) and Π Y (S) S is measure zero. These modifications on the specifications of QE solvers allow us to provide relevant solutions to a wide range of applications, in particular for the stability analysis of numerical schemes. We take advantage of these modifications to design an algorithm handling problems which are out of reach to the best implementations of Cylindrical Algebraic Decomposition. Overview of the algorithm. The pattern of the variant real quantifier elimination algorithm in [20] shares some patterns of the resolution scheme of algorithms based on the critical point method. The core idea is to compute a polynomial B Q[Y] such that the real hypersurface defined by B = 0 contains the boundary of Π Y (S) by avoiding a recursive projection operator as the one in Cylindrical Algebraic Decomposition. This step is the projection step. Once this is done, one can compute a description of a (cell) decomposition of R r V (B), say C 1,..., C N and produce for each cell C i a sample point y i. This is the decomposition step. This can be handled by Cylindrical Algebraic Decomposition [10] or by parametrized version of roadmap algorithms [5, Chapter 16] (this is another motivation to develop roadmap algorithms as in Subsection 1.4). In practice, we have used Cylindrical Algebraic Decomposition. Discriminating which cell C i should be inserted in the output is done by deciding if its sample point y i belongs to Π Y (S) (which consists in deciding the non-emptiness of S Π 1 Y (y i)). This is the lifting step. In practice, this can be handled by Cylindrical Algebraic Decomposition [10] or dedicated algorithms relying on the critical point method which are derived from algorithms presented in the previous sections : recall that the properness of Π Y will imply that S Π 1 Y (y i) is bounded and we have developed a dedicated implementation for handling these situations. Our algorithmic contribution is mainly on the projection step.

52 46 Chapitre 1. Algorithms in real algebraic geometry Projection step. Given a = (a 1,..., a s ) Q s and I = {i 1,..., i l } {1,..., s}, and an infinitesimal ε, we denote by VI,ε a C ε n+r the algebraic variety defined by f 1 = = f p = g i1 a i1 ε = = g il a il ε = 0. Théorème [20, Lemma 9] Let y be a point in the boundary of Π Y (S). There exists I = {i 1,..., i l } with 0 l min(s, n c + 1) such that y Π Y (lim ε 0 crit(π Y, VI,ε a )). Moreover, there exists a non-empty Zariski open set O C s such that for all a O Q s and all I {1,..., s}, F, g i1 a i1 ε,..., g il a il ε Q ε [X, Y] is radical, equidimensional of codimension min(c + l, n + r) and VI,ε a is smooth. As recalled in Subsection 1.2.5, given a polynomial family defining lim ε 0 crit(π Y, VI,ε a ), Gröbner bases allow us to compute algebraic representations of the Zariski closure of Π Y (lim ε 0 crit(π Y, VI,ε a )) using elimination orderings. Thus, it remains to compute algebraic representations of lim ε 0 crit(π Y, VI,ε a ). The result below extends Theorem (see Subsection 1.3.3) and allows us to avoid to perform our computations in Q(ε). Given a matrix J, Minors(J, c, p, l) denotes the set of (c + l, c + l) minors of J obtained by considering c lines among the first p columns. We reuse the notations of Theorem Théorème [20, Lemma 10 and Lemma 11] Suppose that a O Q s and let I = {i 1,..., i l } {1,..., s}. Denote by P I the polynomial family F, g i1,..., g il and set I = Minors(jac(P I, X), c, p, l) and I = Minors(jac(P I, [X, Y]), c, p, l). Consider now the ideal JI a = F, g i2 a i1 g i1 a i2,..., g il a i1 g i1 a il, I. The algebraic variety defined by (JI a : I ) + g i1 equals the Zariski closure of lim ε 0 crit(π Y, VI,ε a ). Moreover, its image by Π Y has co-dimension greater than 1. One can design a projection operator from Theorems and It consists in choosing a = (a 1,..., a s ) Q s and for all I = {i 1,..., i l } {1,..., s} of cardinality min(s, n c + 1) perform the following operations : compute I = Minors(jac(P I, X), c, p, l) and I = Minors(jac(P I, [X, Y]), c, p, l) where P I the polynomial family {F, g i1,..., g il } ; compute a polynomial B I in ((JI a : I ) + g i1 ) Q[Y]. If the choice of a is generic enough, (JI a : I ) Q[Y] is not empty (see [20]). By Theorems and 1.5.4, the product of all the B I s for I {1,..., s} of cardinality min(s, n c + 1) defines a hypersurface containing the boundary of π Y (S).

53 1.5. Towards quantifier elimination 47 FIGURE 1.7 Example for the Variant Quantifier Elimination Algorithm Note that ideal theoretic operations (saturation and elimination) can be done by Gröbner bases (see e.g. [17]). Example. We illustrate the algorithm with Y = [Y ], X = [X 1, X 2 ], F = {f} = {X1 2 + X2 2 1} and G = {g} = {X2 1 Y (X 2 1) 2 }. On the figure aside V (f) is the red cylinder and V (g) is the Whitney umbrella in blue. Suppose that a = (1) O Q. When we enter in the loop with I =, we get B I = {1} since, when I =, I = I on this example. When we enter in the loop with I = {1}, we get I = { 4 X 1 (X X 2 Y )} and I = { 4 X 1 (X X 2 Y ), 2 X 3 2 1, 2 X 2 X } 1. Then, we compute a set of generators of F I : I, obtaining S = { X X 2 2 1, X X 2 Y }. Note that V (S) R 3 is the green curve on the figure above. This curve can be interpreted as the Zariski closure of the algebraic containing crit(π Y, V (f, g e)) for e generic. Finally, we compute a set of generators of S {g} Q[X], obtaining { Y 2}. Degree bounds. Let B Q[Y] be the polynomial computed by the projection operator described above. Providing degree bounds on B is not enough to give complexity estimates but gives a hint on the quality of the algorithm. We set D f = max(deg(f 1 ),..., max(deg(f k )), D g = max(deg(g 1 ),..., deg(g s )) and D = max(d f, D g ). Théorème The hypersurface defined by B = 0 has a degree bounded by D p f Dr min(s,n c+1) i=0 D l g ((c + l)d) n c l. Note that the above bound is singly exponential in the number of variables. When p = c, the above bound can be improved to