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Ubiquity of Lojasiewicz’s example of a nonbasic semialgebraic set.

Andradas Heranz, Carlos and Ruiz Sancho, Jesús María (1994) Ubiquity of Lojasiewicz’s example of a nonbasic semialgebraic set. Michigan Mathematical Journal, 41 (3). pp. 465-472. ISSN 0026-2285

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Abstract

Lojasiewicz pointed out in 1965 that the semialgebraic set {x < 0}[{y < 0} in R2 is not
basic: it is not the solution of a simultaneous system of inequalities. In this example the
Zariski closure of the topological boundary crosses the set. The purpose of the present paper is to show that this is the only obstruction to a set being basic. A semialgebraic
set S contained in a real algebraic set X in Rn is said to be generically basic if there are regular functions f1, . . . , fs, h on X, with h 6= 0, such that
S \ {x 2 X | h(x) = 0} = {x 2 X | f1(x) > 0, . . . , fs(x) > 0} \ {x 2 X | h(x) = 0}.
Let S be the interior of the closure of Int(S) \ Reg(X). The generic Zariski boundary
@ZS of S is defined to be the Zariski closure of Reg(X) \ (S
\S). One says that S is crossed by its generic Zariski boundary when (1) dim(S \ @ZS) = d − 1, and (1) S contains some regular points of @ZS of dimension d − 1.
If Y is an irreducible algebraic set in Rm and f : Y ! X is a birational map, then the semialgebraic set f−1(S) is called a birational model of S. Theorem: A semialgebraic set S is generically basic if and only if no birational model of S is crossed by its generic Zariski boundary.

Item Type:Article
Uncontrolled Keywords:Generically basic semialgebraic set; Birational model
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14840
Deposited On:18 Apr 2012 10:07
Last Modified:06 Feb 2014 10:10

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