  Previously, we have systematically constructed explicit real algebraic
functions which are represented as the compositions of smooth real algebraic
maps whose images are domains surrounded by hypersurfaces of degree 1 or 2 with
canonical projections. Here we give new examples with the hypersurfaces each of
which is the product of a connected component of a hyperbola and a copy of the
$1$-dimensional affine space explicitly. As a related future work we also
discuss problems to obtain the zero sets of some real polynomials explicitly
from increasing sequences of real numbers.
  This is motivated by a problem in theory of smooth functions proposed first
by Sharko: can we construct nice smooth functions whose Reeb graphs are as
prescribed? The Reeb space of a smooth function is the naturally obtained graph
whose underlying space is the quotient space of the manifold consisting of
connected components of preimages. The author first considered variants
respecting the topologies of the preimages and obtained several results before.
Our work is also motivated by real algebraic geometry, pioneered by Nash. We
can know existence of real algebraic structures of smooth manifolds and some
general sets and we already know several approximations of smooth maps by real
algebraic maps. Our interest lies in explicit construction, which is difficult.
