, Let f : Cent X ? R be a real function. The following properties are equivalent: (1) f ? K 0 (Cent X). (2) There exists a resolution of singularities ? : Y ? X

, ) We have ?(Y ) = Cent X by Proposition 1.5. (2) The equality f ? ? ? O(Y ) in the statement of the previous proposition means that there exists g ? O(Y ) such that f ? ? = g on Y, Let ? : Y ? X be a resolution of singularities

C. Andradas, L. Bröcker, and J. M. Ruiz, Constructible sets in real geometry, 1996.
DOI : 10.1007/978-3-642-80024-5

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, 1969.

J. Bochnak, M. Coste, and M. Roy, Real algebraic geometry, 1998.

L. Van-den-dries, Tame topology and o-minimal structures, vol.248, 1998.

D. Eisenbud, Commutative Algebra with a view toward algebraic geometry, Graduate texts in mathematics, vol.150, 2004.

G. Fichou, J. Huisman, F. Mangolte, and J. Monnier, Fonctions régulues, vol.718, pp.103-151, 2016.

G. Fichou, J. Monnier, and R. Quarez, Continuous functions on the plane regular after one blowing-up, Math. Z, vol.285, pp.287-323, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01893248

G. Fichou, J. Monnier, and R. Quarez, Weak and semi normalization in real algebraic geometry, Arxiv, 2018.

J. Kollár, W. Kucharz, and K. Kurdyka, Curve-rational functions, Math. Ann, vol.370, issue.1-2, pp.39-69, 2018.

J. Kollár and K. Nowak, Continuous rational functions on real and p-adic varieties, vol.279, pp.85-97, 2015.

W. Kucharz, Rational maps in real algebraic geometry, Adv. Geom, vol.9, issue.4, pp.517-539, 2009.

W. Kucharz and K. Kurdyka, Stratified-algebraic vector bundles, J. Reine Angew. Math

K. Kurdyka, Ensembles semialgébriques symétriques par arcs, Math. Ann, vol.282, pp.445-462, 1988.

K. Kurdyka and A. Parusi?ski, Arc-symmetric sets and arc-analytic mappings in Arc spaces and additive invariants in real algebraic and analytic geometry, Panor. Synthèses. Soc. Math. France, vol.24, pp.33-67, 2007.

F. Mangolte, Variétés algébriques réelles, vol.24, 2017.

H. Matsumura, Commutative algebra, Cambridge studies in advanced mathematics 8, 1989.

J. Monnier, Semi-algebraic geometry with rational continuous functions, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01286990

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