Abstract
Let X be a real algebraic subset of \({{\mathrm{\mathbb {R}}}}^n\). We investigate the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.
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Acknowledgements
I want to thank G. Fichou and R. Quarez for stimulating conversations concerning regulous functions. I thank also F. Mangolte and D. Naie for interesting and helpful discussions. I thank a referee of an earlier version of the paper for pointing out to us a mistake in this earlier version. In memory of J.-J. Risler.
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Communicated by Jean-Yves Welschinger.
