  We present new real algebraic maps of non-positive codimensions with
prescribed images whose boundaries consist of explicit non-singular real
algebraic hypersurfaces satisfying so-called "transversality" as follows.
Explicit information on important real polynomials is given. Preimages are
one-point sets or products of spheres. They are locally like so-called moment
maps.
  Celebrated theory of Nash and Tognoli says that smooth closed manifolds are
{\it non-singular} real algebraic manifolds and the zero sets of some real
polynomial maps. In general, we can approximate smooth functions or more
generally, maps, by real algebraic ones. It is in general difficult to have
explicit examples. We have constructed maps of a specific class of the present
class containing the canonical projections of the unit spheres previously where
preimages are one-point sets or spheres.
  We also present explicit families of functions represented as compositions of
such maps with the canonical projections.
