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

Let S ( M ) be the ring of (continuous) semialgebraic functions on a semialgebraic set M andS * ( 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 ofS * ( M )at the multiplicative subset W M of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusioni M : 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 ) 2at 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 ) 2is 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 . If such is the case, our procedure allows us to compute the size s of Spec ( j ) 2 − 1( p ) . If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p .