Representing elementary semi-algebraic sets by a few polynomial inequalities: A constructive approach

Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,…,p_s such that P= {x \in R^d : p_1(x) \ge 0, >…, p_s(x) \ge 0 }; in this case p_1,…,p_s are said to represent P. Denote by $n$ the maximal number of the polynomials from {p_1,…,p_s} that vanish in a point of P. If P is non-empty and bounded, we show that it is possible to construct n+1 polynomials representing P. Furthermore, the number n+1 can be reduced to n in the case when the set of points of P in which n polynomials from {p_1,…,p_s} vanish is finite. Analogous statements are also obtained for elementary open semi-algebraic sets.