[2304.02372] Explicit smooth real algebraic functions that may have both compact and non-compact prior images on smooth real algebraic varieties

Download the PDF of Naoki Kitazawa’s paper titled Explicit smooth real algebraic functions that may have both compact and non-compact preimages on smooth real algebraic varieties.

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overview: In our previous work, we constructed explicit smooth real algebraic functions that may have both compact and non-compact prior images on smooth real algebraic varieties. This article introduces a variation of it. Our result is novel in that we obtain unsuitable smooth real algebraic functions on smooth real algebraic varieties that satisfy explicit conditions on the (non-)compactness of the preimage. Until now, manifolds have only been semi-algebraic. Obviously, this primarily contributes to his two different areas of mathematics. One is the application of differentiable maps to singularity theory and differential topology. More precisely, building a smooth map using the required preimages. Another is real algebraic geometry. More precisely, the explicit construction of smooth real algebraic functions and maps, but in quite a few cases we can know the existence of smooth maps by such a class of maps and consider approximations.