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.

Boundary slices and the P-Euler condition

Ghiloni, Riccardo
2007-01-01

Abstract

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.
2007
4
Ghiloni, Riccardo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/69764
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