Globalization and compactness of McCrory–Parusiński conditions

Let X ⊂ R be a closed semialgebraic set and let f P(X) be the ring obtained from the characteristic function of X by the operations +, −, ∗, the half link operator and by the polynomial operations with rational coefficients which preserve finite formal sum of signs. McCrory and Parusiński proved that necessary conditions for X to be semialgebraically homeomorphic to a real algebraic set is that X is P–euler, i.e, all the functions in f P(X) are integer–valued. These conditions are local. In this paper, we give global versions of these conditions. For example, we show that, if X is P–euler and M is a compact Nash submanifold of R transverse to some semialgebraic Whitney stratification of X, then X ∩M is P–euler and, for each φ ∈ f P(X ∩M), the Euler integral of φ is even. Moreover, we have the following result: Let F be a family of closed semialgebraic sets and let Fc be the family of all X ∈ F such that X is compact. Suppose F is closed under the inverse images of regular maps. Then all the elements of F are P–euler if and only if all the elements of Fc are. Since it is known that every compact arc–symmetric semialgebraic set is P–euler, we infer that all the arc– symmetric semialgebraic sets are P–euler, answering affirmatively to a question by Kurdyka, McCrory and Parusiński.