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

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

2008-04-14

arxiv.org/abs/0804.2134v1

Mathematics

Algebraic Geometry

14 pages, 4 figures

Scientific paper

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.

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