Polynomial Maps of Affine Quadrics

Since the article [3] on polynomial maps of spheres appeared about 25 years ago, there have been a number of papers on the theory of rational maps of real varieties (see the references in [2], for example) which have many interesting things to say about the representation of homotopy classes by algebraic maps. As far as I know, however, the problem of representing elements in the homotopy group nn(S ) by polynomial maps of the real sphere S, in the case n even, remains unsolved. In particular, we do not seem to know if there is a polynomial self map of the 2-sphere of Brouwer degree 2. Of course, the case of n odd was settled by Theorem 1 of [3], whose proof demonstrated how to construct a homogeneous polynomial map of S of Brouwer degree k and algebraic degree \k\. We recall that nn(S ) = [S, S] stands for the set of homotopy classes of maps of spheres S -*• S. This set forms a group in a natural way, and there is a suspension homomorphism a: nn(S ) -> nn+1(S ) which, in the case n = k, is an isomorphism for positive integers n and establishes a bijection of nn(S ) with n^S) which in turn is isomorphic to the integers via the winding number. Hence an element of 7rn(.S ) corresponds to an integer which we refer to as the Brouwer degree of the element. From a homotopy point of view, nothing is lost by complexifying the problem. This means considering the affine quadric Q~ defined as the locus of points (z15 ...,2n) in C n satisfying the equation