08-06
Representing elementary semi-algebraic sets by a few polynomial inequaltities: A constructive approach
by Averkov, G.
Preprint series: 08-06, Preprints
- MSC:
- 14P10 Semialgebraic sets and related spaces
- 14Q99 None of the above but in this section
- 03C10 Quantifier elimination and related topics
- 90C26 Nonconvex programming, quasiconvex programming
Abstract: Let P be an elementary closed semi-algebraic set in Rd, i.e., there exist realpolynomials p1,...,ps (s N)such that P = x Rd : p1(x) 0,...,ps(x) 0\xaf; 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 nite. Analogous statements are also obtained for elementary open semi-algebraic sets.
Keywords: Approximation, elementary symmetric function, Lojasiewicz\x92s Inequality, polynomial optimization, semi-algebraic set, Theorem of Broecker and Scheiderer
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