How to test in subexponential time whether two points can be connected by a curve in a semialgebraic set

Abstract

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) ∈ (