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AS a result of the Tarski-Seidenberg theorem, problems in real algebraic geometry usually have constructive solutions. In this article we show that this is not always the case. We consider the following problem, which is of interest for its own sake. Let S" c R" +1 be a nonsingular compact algebraic surface of degree of. Let S n be isotopic to the standard hypersphere S n C R w+1 . It is well known that it is not always possible to connect Z" and S n by an isotopy passing via nonsingulardoi:10.1090/s0273-0979-1989-15698-x fatcat:s5pbvemwibcy7etowwkqza2wtm