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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 connecteddoi:10.1006/jcom.1997.0434 fatcat:pexulakn4bcylhdaruqmg3iryu