Explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds

In our previous work, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds. This paper presents its variant. Our result is new in obtaining non-proper smooth real algebraic functions on smooth real algebraic manifolds satisfying explicit conditions on (non-)compactness of preimages whereas previously the manifolds are only semi-algebraic. Explicitly, this mainly contributes to two different regions of mathematics. One is singularity theory of differentiable maps and applications to differential topology. More precisely, construction of nice smooth maps with desired preimages. The other is real algebraic geometry. More precisely, explicit construction of smooth real algebraic functions and maps whereas we can know the existence and consider approximations of smooth maps by maps of such classes in considerable cases.