On Computing a Set of Points Meeting Every Cell Defined by a Family of Polynomials on a Variety

We consider a family of s polynomials, P = fP ; . . . ; P g; in k variables with coefficients in a real closed field R; each of degree at most d; and an algebraic variety V of real dimension k which is defined as the zero set of a polynomial Q of degree at most d. The number of semi-algebraically connected components of all non-empty sign conditions on P over V is bounded by s (O(d)) . In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non-empty sign condition of P over V . Its complexity is s d . This interpolates a sequence of results between the Ben-Or–Kozen–Reif algorithm which is the case k = 0, in one variable, and the Basu–Pollack–Roy algorithm which is the case k = k. It improves the results where the same problem was solved in time s d . ©1997 Academic Press