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
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用一些多项式不等式表示初等半代数集：一种构造性方法
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发表时间：
2008
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通讯作者：
G. Averkov
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作者：
G. Averkov
设P是R^d中的初等闭半代数集，即，存在真实的多项式p_1，.，p_s使得P= {x in R^d：p_1（x）ge 0，>.，p_s（x）ge 0 };在这种情况下，p_1，.，p_s被称为表示P。用$n$表示来自{p_1，.，若P是非空有界的，我们证明了构造n+1个多项式表示P是可能的。此外，当P的点集中有n个多项式从{p_1，...，p_s} vanish是有限的。对于初等开半代数集也得到了类似的结论。
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.