Is there an algorithm which, given two polynomials in $n$ variables with real coefficients, $p(x)$, and $q(x)$, will determine whether the zero sets $p^{-1}(0), q^{-1}(0)\subset R^n$, are homeomorphic to each other?


(also same question for polynomials over $C$ with $R^n$ replaced by $C^n$).



Determining if two algebraic sets are homeomorphic 
I think the answer to the "real" version of the question is no. Here are some remarks.


1. One can realize each smooth manifold as a real algebraic variety in a Euclidean space. So one can realize each smooth compact manifold as the zero set of a single polynomial, by taking the sum of the squares of the polynomials that generate the ideal of the corresponding algebraic variety.
2. In dimensions $\leq 7$ every PL-manifold admits a smooth structure. So given a PL-manifold one can "smoothen" it and construct a homeomorphic real algebraic variety. The question is whether this can be done constructively (and moreover so that the resulting variety is defined over the rationals, if one considers the "rational" version). This looks plausible. If it is true then one can use the fact that the homeomorphism problem for PL-manifolds of dimension 4 is undecidable.


