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Boundary slices and the 𝒫‐Euler condition

Let X⊂ℝn be a closed semialgebraic set, and let 𝒫̃(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 𝒫‐Euler; that is, all the functions in 𝒫̃(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 Parusiński.