Zeroes of polynomials on definable hypersurfaces: Pathologies exist, but they are rare

Given a sequence {Zd}d?N of smooth and compact hypersurfaces in Rn-1, we prove that (up to extracting subsequences) there exists a regular definable hypersurface ? RPn such that each manifold Zd is diffeomorphic to a component of the zero set on of some polynomial of degree d. (This is in sharp contrast with the case when is semialgebraic, where for example the homological complexity of the zero set of a polynomial p on is bounded by a polynomial in deg(p).) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface ? RPn containing a subset D homeomorphic to a disk, and a family of polynomials {pm}m?N of degree deg(pm) = dm such that (D, Z(pm)nD) ~ (Rn-1, Zdm ), i.e. the zero set of pm in D is isotopic to Zdm in Rn-1. This says that, up to extracting subsequences, the intersection of with a hypersurface of degree d can be as complicated as we want. We call these 'pathological examples'. In particular, we show that for every 0 = k = n - 2 and every sequence of natural numbers a = {ad}d?N there is a regular, compact semianalytic hypersurface ? RPn, a subsequence {adm }m?N and homogeneous polynomials {pm}m?N of degree deg(pm) = dm such that bk( n Z(pm)) = adm . (0.1) (Here bk denotes the kth Betti number.) This generalizes a result of Gwozdziewicz et al. [13]. On the other hand, for a given definable we show that the Fubini-Study measure, in the Gaussian probability space of polynomials of degree d, of the set dm,a, of polynomials verifying (0.1) is positive, but there exists a constant c such that