COMPUTING ROADMAPS OF SEMI-ALGEBRAIC SETS ON A VARIETY

Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1, . . . , Xk] and S a semi-algebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0, P > 0, P = 0 with P ∈ P , where P is a finite subset of R[X1, . . . , Xk]. A semi-algebraic set C is semi-algebraically connected if it is non-empty and is not the union of two non-empty disjoint semi-algebraic sets which are closed and open in C. A semi-algebraically connected component of S is a semi-algebraic subset of S which is semi-algebraically connected, and closed and open in S . Semi-algebraic sets have a finite number of semi-algebraically connected components ([5], page 34). A roadmap of S, which we denote R(S), is a semi-algebraic set of dimension at most one contained in S which satisfies the roadmap conditions: RM1 For every semi-algebraically connected component C of S, C ∩R(S) is semialgebraically connected. RM2 For every x ∈ R, and for every semi-algebraically connected component C′ of Sx, C ′ ∩R(S) 6= ∅. Here, and everywhere else in this paper, π is the projection on the first coordinate and for X ⊂ R, SX is S ∩ π−1(X). We also use the abbreviations Sx, S<c, and S≤c for S{x}, S(−∞,c), and S(−∞,c] respectively. Algorithms for the construction of roadmaps are described in terms of the parameters k, k′, s, d where k is the dimension of the ambient space, k′ is the dimension of Z(Q), s is the number of polynomials in P and d is a bound on the degrees of the polynomials in P and the polynomial Q. Given a roadmap R(S) and a point p ∈ S the connecting subroutine outputs a semi-algebraic continuous path in Sπ(p) connecting p to R(S). The connecting subroutine is described in terms of the parameters k, k′, s, d as before and τ which is a bound on the degrees of the polynomials defining p (see section 4 for a discussion of how points are described by polynomials).