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2008-06

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

by Averkov, G..


Series: 2008-06, Preprints

MSC:
14P10 Semialgebraic sets and related spaces
14Q99 None of the above, but in this section
03C10 Quantifier elimination, model completeness and related topics
90C26 Nonconvex programming, global optimization

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¯; 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.

Keywords:
Approximation, elementary symmetric function, Lojasiewicz’s Inequality, polynomial optimization, semi-algebraic set, Theorem of Broecker and Scheiderer