Nonrecursive functions in real algebraic geometry

Alexander Nabutovsky
1989 Bulletin of the American Mathematical Society  
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 nonsingular
more » ... nonsingular algebraic hypersurfaces of degree not higher than d. We prove that it is always possible to connect £ n and S n by an isotopy passing via algebraic surfaces of some degree A which depends on n and d only. Consider for each n the smallest of such degrees as a function A n (d). What can be said about these functions? We prove that for each n > 5, the function A n (d) cannot be majorized by a recursive function of d. Also, some generalizations of these results are stated below.
doi:10.1090/s0273-0979-1989-15698-x fatcat:s5pbvemwibcy7etowwkqza2wtm