Ruiz Sancho, Jesús María2023-06-212023-06-211984G. W. BRUMFIEL, Some open problems, in Ordered fields and real algebraic geometry. Contemporary Math. 8, Amer. Math. Soc. (1982). S. LOJASIEWICZ, Ensembles semi-analytiques. Lecture Notes 1965 at I.H.E.S., Bures-sur-Yvette; reproduit n. 466.765, Ecole Polythecnique, Paris. T. MOSTOWSKI, Some properties of the ring of Nash functions. Ann. Scuola Norm. Sup. Pisa, III, 243-266 (1976). J. J. RISLER, Sur le théoreme des fonctions composées différentiables. Ann. Institut Fourier, 32 (2), 229-260 (1982). J. M. RUIZ, Central orderings in fields of real meromorphic function germs. Manuscripta Math. 46, 1-3, 193-214 (1984).0003-889X10.1007/BF01193850https://hdl.handle.net/20.500.14352/64767The author proves the following theorem: Let A0 be a closed 1-dimensional semianalytic germ at the origin 0∈Rn. Let Z be a semianalytic set in Rn whose germ Z0 at 0 is closed and A0∩Z0={0}. Then there exists a polynomial h∈R[x1,⋯,xn] such that h∣Z∖{0}>0 and h∣A0∖{0}<0. The proof is by induction on the number of blowing-ups needed to "solve" the set A0. Some implications are then given, in particular a similar result for semialgebraic sets in Rn and polynomials.engjournal articlehttp://link.springer.com/content/pdf/10.1007%2FBF01193850http://www.springer.comrestricted access512.7515.171.5Separation problemsemianalytic germanalytic function germsemialgebraic setsGeometria algebraica1201.01 Geometría Algebraica