Boundary slices and the 𝒫‐Euler condition

Let X ⊂ℝ n be a closed semialgebraic set, and let ̃( 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 Parusiński proved that a necessary condition for X to be homeomorphic to a real algebraic set is that X is ‐Euler; that is, all the functions in ̃( 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 ‐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 ‐Euler, answering affirmatively a question of Kurdyka, McCrory and Parusiński.