Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres

Abstract: Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth surjection onto a manifold whose preimages are diffeomorphic to a unit sphere in the interior of the manifold and single point sets on the boundary with a smooth immersion of codimension $0$. This paper constructs real algebraic maps topologically special generic maps whose images are smoothly embedded manifolds. We are also interested in construction of explicit and meaningful smooth maps in differential topology and recently ones in real algebraic geometry. This has been an important and difficult problem. In such stories, we have previously constructed real algebraic maps topologically regarded as special generic maps. This paper is a kind of additional short remark on such maps.