Boundary slices and the P-Euler condition

Let X be a closed semialgebraic subset of R^n, and let ˜P(X) be the ring obtained from the characteristic function of X by the operations +, −, ∗ and the half link operator, and by the polynomial operations with rational coefficients which preserve finite formal sums of signs. McCrory and Parusinski proved that a necessary condition for X to be homeomorphic to a real algebraic set is that X is P-Euler; that is, all the functions in ˜P(X) are integer-valued. In this paper, we introduce a class of subsets of X, called boundary slices of X. We establish a relation between these subsets of X and the P-Euler condition on X, and we give some applications of this relation. As a consequence, we infer that all the arc-symmetric semialgebraic sets and all the real analytic sets are P-Euler, answering affirmatively a question of Kurdyka, McCrory and Parusinski.