How to test in subexponential time whether two points can be connected by a curve in a semialgebraic set
Authors: D. Y. Grigoriev
Venue: Proceedings of the international symposium on Symbolic and algebraic computation (1990)

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 &sup; F its uniquely defined real closure. In the sequel we consider input polynomials over the ordered ring Zm Z[&dgr;1, &hellip;, &dgr;m] &sub; Qm = Q(&dgr;1, &hellip;, &dgr;m), where &dgr;1, &hellip;, &dgr;m are algebraically independent elements over Q and the ordering in the field Qm is defined as follows. The element &dgr;1 is infinitesimal with respect to Q (i. e. 0 < &dgr;1 < &agr; for any rational number 0 < &agr; &isin; Q) and for each 1 &le; i < m the element &dgr;i+1 > 0 is infinitesimal with respect to the field Qi (cf. [GV 88], [Gr 88]). Thus, let an input quantifier-free formula &Xgr; for the first-order theory of real closed fields be given, containing atomic subformulae of the form &fnof;i &ge; 0, 1 &le; i &le; k where &fnof;i &isin; Zm[X1, &hellip;,Xn]. Any rational function g &isin; Qm(Y1, &hellip;, Y3) can be represented as g = g1/g2 where the polynomials g1, g2 &isin; Zm[Y1, &hellip;, 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, &hellip;, Y3, &dgr;1, &hellip;,&dgr;m). In the sequel we assume that the following bounds are valid: degx1, &hellip;, xn(&fnof;i) < d, deg&dgr;1, &hellip;, &dgr;m (&fnof;i) < d0, l(&fnof;i) &le; M, 1 &le; i &le; k (1) where d, d0, M are some integers. Then the bit-length of the formula &minusb; 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(&Ogr;(n)) time. We use the notation h1 &le; P(h2, &hellip;, ht) for the functions h1 > 0, &hellip;, ht > 0 if for the suitable integers c, &ggr; the inequality h1 &le; c(h2 &middot;&hellip;&middot; ht)&ggr; is fulfilled. Recall that a semialgebraic set (in Fn where F is a real closed field) is a set {II} &sub; 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 &ge; 0) where the polynomials g &isin; F[X1, &hellip;, Xn]. A semialgebraic set {&Xgr;} &sub; (Qm)n is (uniquely) decomposable in a union of a finite number of connected components {&minusb;} = &Cup;1&le;i&le;t {&minusb;i}, each of them in its turn being a semialgebraic set determined by appropriate quantifier-free formula &minusb;i 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 &le; (kd)&Ogr;(n) (see e. g. [GV 88], [Gr 88]). We use the following way of representing the points u = (u1, &hellip;, un) &isin; (@@@@m)n (cf. [GV 88]). Firstly, for the field Qm (u1, &hellip;, un) a primitive element &eegr; is produced such that Qm (u1, &hellip;, un) = Qm[&eegr;], herewith a minimal polynomial @@@@(Z) &isin; Qm[Z] for &eegr; is indicated, furthermore &eegr; = &Sgr;1&le;i&le;n &agr;ivi for some integers 0 &le; &agr;1, &hellip;, &agr;n &le; degZ(@@@@). Also the expressions ui = &Sgr;0&le;j<degz(@@@@) &bgr;(j)i &eegr;j are yielded, where &bgr;(j)i &isin; Qm. Secondly, for specifying the root &eegr; of the polynomial @@@@ a sequence of the signs of the derivatives of all orders @@@@&prime;(&eegr;), @@@@(2)(&eegr;), &hellip;, @@@@(deg(@@@@)) (&eegr;) of the polynomial @@@@ in the point &eegr; is given. Thom's Lemma (see e. g. [FGM 88]) entails that the latter condition uniquely determines the root &eegr; of @@@@. We say that a point u satisfies (D, D0, M)-bound if the following inequalities hold: degZ(@@@@) < D; deg&dgr;1 , &hellip;, &dgr;m(@@@@), deg&dgr;1 , &hellip;, &dgr;m (&bgr;(j)i) &le; D0; l (@@@@), l(&bgr;(j)i) &le; M Then the bit-length of the representation of the point u does not exceed P(M, D, Dm0, n) (cf. [GV 88], [Gr 88]). The main purpose of the paper is to prove the following theorem (see also [VG 91]).