RT Journal Article
T1 On the connected components of a global semianalytic set
A1 Ruiz Sancho, Jesús María
AB Let M be a real analytic manifold and O(M) its ring of global analytic functions. A global semianalytic subset of M is any set Z of the form Z=⋃ i=0  r {x∈M:fi1(x)>0,⋯,fis(x)>0,gi(x)=0}, (1.1), where fij,gi∈O(M). This imitates the definitions of semialgebraic sets and semianalytic germs, and gives rise to the same old basic problems: Can the gi's in (1.1) be omitted if Z is open? Is the closure of Z global semianalytic when Z itself is? And the connected components of Z? In an earlier paper [in Algèbre, 84–95, Univ. Rennes I, Rennes, 1986] we showed that this is possible for the first two questions in case M is compact: our method relied upon the theory of the real spectrum. In this note we deal with the third question and prove Theorem 1.2: Let Z be a global semianalytic subset of a real analytic manifold M. Assume that Z is relatively compact. Then the connected components of Z are global semianalytic subsets of M. For the proof, we use again the real spectrum, plus the solution by Ch. Rotthaus of M. Artin's conjecture on the approximation property of excellent rings [Rotthaus, Invent. Math. 88 (1987), no. 1, 39–63].
PB Walter de Gruyter & co
SN 0075-4102
YR 1988
FD 1988
LK https://hdl.handle.net/20.500.14352/58337
UL https://hdl.handle.net/20.500.14352/58337
DS Docta Complutense
RD 8 abr 2025