Semi-Algebraic Sets and the Limit of the Discounted Value

The article presents a study on a class of polynomial optimization problems over (noncompact) semi-algebraic sets which, by making changes of variables via suitable monomial mappings, become polynomial optimization problems over compact semi-algebraic feasible sets. It is known that the polynomial optimization problems on semi-algebraic feasible sets are satisfactory when the feasible sets are compact. Furthermore, determining whether a polynomial is bounded on such a semi-algebraic set can be replaced by checking whether its support lies in a closed and convex cone corresponding to the semi-algebraic set.

We prove in this chapter a classical result: the number of connected components of a plane section \(\mathrm{P}\cap \mathrm{A}\) of a semialgebraic set \(\mathrm{A}\) is uniformly bounded with respect to \(\mathrm{P}\). An explicit bound is given in terms of the diagram of \(\mathrm{A}\) and the dimension of \(\mathrm{P}\). We give a construction which provides a semialgebraic section of bounded complexity for any polynomial mapping of semialgebraic sets. In particular, any two points in a connected semialgebraic set can be joined by a semialgebraic curve of bounded complexity. We also give the definition of an o-minimal structure on the real field and show that in such a category the uniform bound for the number of connected components of plane sections holds.

Semi-algebraic sets and semi-algebraic functions are essential to specify and certify cylindrical algebraic decomposition algorithms. We formally define in Coq the base operations on semi-algebraic sets and functions using embedded first-order formulae over the language of real closed fields, and we prove the correctness of their geometrical interpretation. In doing so, we exploit a previous formalisation of quantifier elimination on such embedded formulae to guarantee the decidability of several first-order properties and keep our development constructive. We also exploit it to formalise formulae substitution without having to handle bound variables.

In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λa›a∈ℝ, all the bounded semi-algebraic (that is, -definable) sets are definable. Said differently, every such is an expansion of = ‹ℝ, +, <, λa, Bi›a∈ℝ, i∈I where {Bi}i∈I is the collection of all bounded semialgebraic sets and the λa's are scalar multiplication by a. In [PSS] (see Theorem 1.2 below) it was shown that the structure is a proper reduct of ; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that is the only reduct properly between ℳ and . As a first step towards this result, we investigate in this paper the definable sets in reducts such as . (We point out that ‘definable’ will always mean ‘definable with parameters’.)Definition 1.1. Let X ⊆ ℝn. X is called semi-bounded if it is definable in the structure ‹ℝ, +, <, λa, B1, …, Bk›a∈ℝ, where the Bi's are bounded subsets of ℝn.The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the “stretching” of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power.The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in §4.

A subexponential-time algorithm is designed which finds the number of connected components of a semi-algebraic set given by a quantifier-free formula of the first-order theory of real closed fields (for a rather wide class of real close fields, cf. [GV 88], [Gr 88]). Moreover, the algorithm allows for any two points from the semi-algebraic set to test, whether they belong to the same connected component. Decidability of the mentioned problems follows from the quantifier elimination method in the first-order theory of real closed fields, described for the first time by A. Tarski ([Ta 51]). However, complexity bound of this method is nonelementary, in particular, one cannot estimate it by any finite iteration of the exponential function. G. Collins ([Co 75]) has proposed a construction of cylindrical algebraic decomposition, which allows to solve these problems in exponential time. For an arbitrary ordered field F we denote by F ⊃ F its uniquely defined real closure. In the sequel we consider input polynomials over the ordered ring Zm Z[δ1, …, δm] ⊂ Qm = Q(δ1, …, δm), where δ1, …, δm are algebraically independent elements over Q and the ordering in the field Qm is defined as follows. The element δ1 is infinitesimal with respect to Q (i. e. 0 1 i+1 > 0 is infinitesimal with respect to the field Qi (cf. [GV 88], [Gr 88]). Thus, let an input quantifier-free formula X for the first-order theory of real closed fields be given, containing atomic subformulae of the form ƒi ≥ 0, 1 ≤ i ≤ k where ƒi ∈ Zm[X1, …,Xn]. Any rational function g ∈ Qm(Y1, …, Y3) can be represented as g = g1/g2 where the polynomials g1, g2 ∈ Zm[Y1, …, Y3] are reciprocately prime. Denote by l(g) the maximum of bit-lengths of the (integer) coefficients of the polynomials g1, g2 (in the variables Y1, …, Y3, δ1, …,δm). In the sequel we assume that the following bounds are valid: degx1, …, xn(ƒi) δ1, …, δm (ƒi) 0, l(ƒi) ≤ M, 1 ≤ i ≤ k (1) where d, d0, M are some integers. Then the bit-length of the formula m can be estimated by the value L = k M dn dm0 (cf. [CG 83], [Gr 86]). Note that in the case m = 0, i. e. for the polynomials with integer coefficients, the algorithms from [Co 75] allow to produce the connected components (in particular to solve the problems considered in the present paper) within polynomial in M (kd)2(O(n)) time. We use the notation h1 ≤ P(h2, …, ht) for the functions h1 > 0, …, ht > 0 if for the suitable integers c, γ the inequality h1 ≤ c(h2 ·…· ht)γ is fulfilled. Recall that a semialgebraic set (in Fn where F is a real closed field) is a set {II} ⊂ Fn of all points satisfying a certain quantifier-free formula II of the first-order theory of the field F with the atomic subformulae of the form (g ≥ 0) where the polynomials g ∈ F[X1, …, Xn]. A semialgebraic set {X} ⊂ (Qm)n is (uniquely) decomposable in a union of a finite number of connected components {m} = C1≤i≤t {mi}, each of them in its turn being a semialgebraic set determined by appropriate quantifier-free formula mi of the first-order theory of the field Qm (see e. g. [Co 75] for the field F = R, for an arbitrary real closed field one can involve Tarski ([Ta 51]). Note that t ≤ (kd)O(n) (see e. g. [GV 88], [Gr 88]). We use the following way of representing the points u = (u1, …, un) ∈ (

This chapter deals with semi-algebraic sets over a real closed field R. These are the sets defined by a boolean combination of polynomial equations and inequalities. This class of sets has a remarkable property: stability under projection. Several applications of this basic property are investigated. The study of semi-algebraic sets is based mainly on the ∜slicingℝ technique, which makes it possible to decompose them into a finite number of subsets semi-algebraically homeomorphic to open hypercubes. Using this decomposition, we show that a semi-algebraic set has a finite number of semi-algebraically connected components. The notions of connectedness and compactness over a real closed field, other than ℝ, require some care. Nevertheless, closed and bounded semi-algebraic subsets of R n preserve several of the properties known in the case R = ℝ. They are proved using the curve-selection lemma. All this is the subject of the first five sections of this chapter. In Section 6, we study continuous semi-algebraic functions and we show Łojasiewicz’s inequality. Section 7 deals with the separation of disjoint closed semi-algebraic sets. Section 8 introduces the notion of dimension for a semi-algebraic set and establishes its expected properties. Finally, the last section contains essentially an implicit function theorem in the semi-algebraic framework (this result is well known over R but it is also useful over real closed fields other than ℝ).

In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326. MR 2156215 (2006k:14108)], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.

Surjective Nash maps between semialgebraic sets

QEPCAD B 2 is a system for computing with semi-algebraic sets. a semi-algebraic set is a subset of ℝ n that can be defined as the set of points satisfying a boolean formula combining polynomial equalities and inequalities in the variables x 1 ,..., x n . So, for example, the upper-right quadrant of the unit disk is a semi-algebraic set, since it has the defining formula[see pdf for formula]Many important problems in mathematics, science and engineering boil down to questions about semi-algebraic sets. QEPCAD B allows its users to compute with semi-algebraic sets specified by defining formulae. Computation is exact and symbolic, results being returned in the same language of defining formulae. The basic operations the system supports are formula simplification and quantifier elimination Quantifier Elimination: Adding quantifiers to a defining formula is, in a sense, asking a question. For example, ∃ x [ x 2 + bx + c =0] is the question "when does x 2 + bx + c have a real root?" The well-known answer "when b 2 −4 c ≥0" is an equivalent formula from which the quantified variable has been eliminated. Quantifier elimination algorithms, which produce such equivalent formulae, can be seen as providing "answers" to "questions" about semi-algebraic sets. Formula Simplification: Many procedures in mathematics, performed both manually and mechanically, produce "answers" in the form of defining formulae. These defining formulae are often not particularly nice characterizations of the sets they define--hence the need for formula simplification. For example, QEPCAD B determines that the formula F :=1+ b 2 − c 2 ≥ b ∧− c ( b 2 − c 2 ) 3 +3 b 2 c ( b 2 − c 2 )∨ b 2 − c 2 < b under the assumption b > 0 ∧ c > 0 ∧ 1 < b + c ∧ b < 1 + c ∧ c < 1 = b is equivalent to F ′ := c 2 − bc − 1 > 0. Obviously, F ′ was a considerably better characterization for subsequent computations in the application from which this arose. Cylindrical Algebraic Decomposition (CAD): A CAD is essentially a data-structure providing an explicit representation of a semi-algebraic set. This representation is expensive to compute, but it contains so much information about the set it represents that quantifier elimination and simplification are easily accomplished, which is why CAD is the basis for these operations in QEPCAD B. A little insight into what CAD is and how it is used is provided by the following figures, produced by QEPCAD B, which show the CAD representation for the formula F from the simplification example, followed by the CAD representing F restricted by the given assumptions, followed by the simplified CAD representation of the same set, which is what was used to provide the simplified output formula F ′.This exhibit focuses on using QEPCAD B as a problem-solving tool. Examples trace problems from the application areas from which they arise, through the initial formulation of quantifier elimination or formula simplification problems, through refining problem formulations to take advantage of QEPCAD B's strengths and avoid its weaknesses. Limitations of the system and tradeoffs versus other tools are also discussed.

Rectilinearization of semi-algebraic p-adic sets and Denef's rationality of Poincaré series

Quantitative results on quadratic semi-algebraic sets

The moment problem for compact semi-algebraic sets has been solved in [S1] (see [PD], Section 6.4, for a refinement of the result). In the terminology explained below, this means that each defining sequence f of a compact semialgebraic set Kf has property (SMP ). On the other hand, for many noncompact semi-algebraic sets (for instance, sets containing a cone of dimension two [KM], [PS]) the moment problem is not solvable. Only very few noncompact semi-algebraic sets (classes of real algebraic curves [So], [KM], [PS] and cylinder sets [Mc]) are known to have a positive solution of the moment problem. In this paper we study semi-algebraic setsKf such that there exist polynomials h1, . . ., hn which are bounded on the set Kf . Our main result (Theorem 1) reduces the moment problem for the set Kf to the moment problem for the “fiber sets” Kf ∩Cλ, where Cλ is real algebraic variety {x ∈ R : h1(x) = λ1, . . ., hn(x) = λn}. From this theorem new classes of non-compact closed semi-algebraic sets are obtained for which the moment problem has an affirmative solution. Combined with a result of V. Powers and C. Scheiderer [PS], it follows that tube sets around certain real algebraic curves have property (SMP ) (see Theorem 9). Let f = (f1, . . ., fk) be a finite set of polynomials fj ∈ R[x] ≡ R[x1, . . ., xd] and let K ≡ Kf be the associated closed semi-algebraic subset defined by

A semialgebraic set is a subset of real space defined by polynomial equations and inequalities. A semialgebraic set is a union of finitely many maximally connected components. In this talk, we consider the problem of deciding whether two given points in a semialgebraic set are connected, that is, whether the two points lie in the same connected component. In particular, we consider the semialgebraic set defined by f not equal 0 where f is a given bivariate polynomial. The motivation comes from the observation that many important/non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to its importance, there has been intense research effort on the problem. We will describe a method based on gradient fields and provide a sketch of the proof of correctness based on the Morse complex. The method seems to be more efficient than the previous methods in practice.

For any r≥1 and any i, 0≤i≤r, an i-dimensional cell (in Er) is a subset of r-dimensional Euclidean space Er homeomorphic to the i-dimensional open unit ball. A subset of Er is said to possess a cellular decomposition (c.d.) if it is the disjoint union of finitely many cells (of various dimensions). A semialgebraic set S (in Er) is the set of all points of Er satisfying some given finite boolean combination φ of polynomial equations and inequalities in r variables. φ is called a defining formula for S. A real algebraic variety, i.e. the set of zeros in Er of a system of polynomial equations in r variables, is a particular example of a semialgebraic set. It has been known for at least fifty years that any semialgebraic set possesses a c.d., but the proofs of this fact have been nonconstructive. Recently it has been noted that G. E. Collins' 1973 quantifier elimination algorithm for the elementary theory of real closed fields contains an algorithm for determining a c.d. of a semialgebraic set S given by its defining formula, apparently the first such algorithm. Specifically, each cell c of the c.d. C of S is itself a semialgebraic set, and for every c in C, a defining formula for c and a particular point of c are produced. In the present paper we provide a proof of this fact, our proof amounting to a description of Collins' algorithm from a theoretical point of view. We then show that the algorithm can be extended to determine the dimension of each cell in a c.d. and the incidences among cells. A computer implementation of the algorithm is in progress.KeywordsAtomic FormulaQuantifier EliminationReal Algebraic VarietyCellular DecompositionReal Closed FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Most of the curves and surfaces encountered in geometric modelling are defined as the set of solutions of a system of algebraic equations and inequalities (semi-algebraic sets). Many problems from different fields involve proximity queries like finding the (nearest) neighbours or quantifying the neighbourliness of two objects. The Voronoi diagram of a set of sites is a decomposition of space into proximal regions. The proximal region of a site is the locus of points closer to that site than to any other one. Voronoi diagrams allow one to answer proximity queries after locating a query point in the Voronoi zone it belongs to. The dual graph of the Voronoi diagram is called the Delaunay graph. Only approximations by conics can guarantee a proper order of continuity at contact points, which is necessary for guaranteeing the exactness of the Delaunay graph. The theoretical purpose of this thesis is to elucidate the basic algebraic and geometric properties of the offset to an algebraic curve and to reduce the semi-algebraic computation of the Delaunay graph to eigenvalues computations. The practical objective of this thesis is the certified computation of the Delaunay graph for low degree semi-algebraic sets embedded in the Euclidean plane. The methodology combines interval analysis and computational algebraic geometry. The central idea of this thesis is that a (one time) symbolic preprocessing may accelerate the certified numerical evaluation of the Delaunay graph conflict locator. The symbolic preprocessing is the computation of the implicit equation of the generalised offset to conics. The reduction of the Delaunay graph conflict locator for conics from a semi-algebraic problem to a linear algebra problem has been possible through the use of the generalised Voronoi vertex (a concept introduced in this thesis). The certified numerical computation of the Delaunay graph has been possible by using an interval analysis based library for solving zero-dimensional systems of equations and inequalities (ALIAS). The certified computation of the Delaunay graph relies on theorems on the uniqueness of a root in given intervals (Kantorovitch, Moore-Krawczyk). For conics, the computations get much faster by considering only the implicit equations of the generalised offsets.