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In our previous work, we have constructed explicit smooth real algebraicfunctions which may have both compact and non-compact preimages on smooth realalgebraic manifolds. This paper presents its variant. Our result is new inobtaining non-proper smooth real algebraic functions on smooth real algebraicmanifolds satisfying explicit conditions on (non-)compactness of preimageswhereas previously the manifolds are only semi-algebraic. Explicitly, thismainly contributes to two different regions of mathematics. One is singularitytheory of differentiable maps and applications to differential topology. Moreprecisely, construction of nice smooth maps with desired preimages. The otheris real algebraic geometry. More precisely, explicit construction of smoothreal algebraic functions and maps whereas we can know the existence andconsider approximations of smooth maps by maps of such classes in considerablecases.

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