On the size of the fibers of spectral maps induced by semialgebraic embeddings

الأصل · EN

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M⊂Rᵐ and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)₁: Spec(S(N))→ Spec(S(M)) and Spec(j)₂: Spec(S*(N))→ Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset Wₘ of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iₘ: Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)₂: Spec(S*(N))→ Spec(S*(M)). If we denote Z:= cl Spec(S*(M))(M N), it holds that the restriction map Spec(j)₂|: Spec(S*(N)) Spec(j)₂⁻¹(Z)→ Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)₂ at the points of Z. The size of the fibers of prime ideals `close' to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)₂ is surjective and the generic fiber of a prime ideal p∈ Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)₂⁻¹(p) is a finite set for p∈ Z.