RT Journal Article
T1 Ubiquity of  Lojasiewicz’s example of a nonbasic semialgebraic set.
A1 Andradas Heranz, Carlos
A1 Ruiz Sancho, Jesús María
AB Lojasiewicz pointed out in 1965 that the semialgebraic set {x < 0}[{y < 0} in R2 is notbasic: it is not the solution of a simultaneous system of inequalities. In this example theZariski 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 semialgebraicset 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 thatS \ {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.
PB Michigan Mathematical Journal
SN 0026-2285
YR 1994
FD 1994
LK https://hdl.handle.net/20.500.14352/57171
UL https://hdl.handle.net/20.500.14352/57171
LA eng
NO DGICYT
DS Docta Complutense
RD 10 abr 2025