Representing Elementary Semi-Algebraic Sets by a Few Polynomial Inequalities: A Constructive Approach
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چکیده
Let P be an elementary closed semi-algebraic set in R, i.e., there exist real polynomials p1, . . . , ps (s ∈ N) such that P = ̆ x ∈ R : p1(x) ≥ 0, . . . , ps(x) ≥ 0 ̄ ; in this case p1, . . . , ps are said to represent P . Denote by n the maximal number of the polynomials from {p1, . . . , ps} 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 {p1, . . . , ps} vanish is finite. Analogous statements are also obtained for elementary open semi-algebraic sets. 2000 Mathematics Subject Classification. Primary: 14P10, Secondary: 14Q99, 03C10, 90C26
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تاریخ انتشار 2008