  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.
