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Let P be an elementary closed semi-algebraic set in R^d, i.e., there existreal 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 apoint of P. If P is non-empty and bounded, we show that it is possible toconstruct n+1 polynomials representing P. Furthermore, the number n+1 can bereduced to n in the case when the set of points of P in which n polynomialsfrom \{p_1,...,p_s\} vanish is finite. Analogous statements are also obtainedfor elementary open semi-algebraic sets.

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