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 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:=ClSpec(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. 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. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The merger history of massive spheroids since z ∼ 1 is size independent

Using a compilation of 379 massive (stellar mass M ≳ 10^11 M_⊙) spheroid-like galaxies from the near-infrared Palomar/DEEP-2 survey, we investigated, up to z ∼ 1, whether the presence of companions depends on the size of the host galaxy. We explored the presence of companions for mass ratios with respect to the central massive galaxy down to 1 : 10 and 1 : 100, and within projected distances of 30, 50 and 100 kpc of these objects. We found evidence that these companions are equally distributed around both compact and extended massive spheroid-like galaxies. This suggests that, at least since z ∼ 1, the merger activity in these objects is nearly homogeneous across the whole population and that the merger history is not affected by the size of the host galaxy. Our results could indicate that compact and extended massive spheroid-like galaxies are increasing in size at the same rate.