Construction of real algebraic functions with prescribed preimages

Nash and Tognoli show that smooth closed manifolds can be the zero sets of some real polynomial maps and non-singular. The canonical projections of spheres naturally embedded in the $1$-dimensional higher Euclidean spaces and some natural functions on projective spaces, Lie groups and their quotient spaces are important examples of real algebraic functions being also Morse. In general, it is difficult to construct such examples of maps and the structures of the manifolds. In addition the maps are hard to understand globally. We construct examples by answering to a problem from singularity theory and differential topology. It asks whether we can reconstruct nice smooth functions with prescribed preimages. We have previously given an answer with real algebraic functions. This previous result is one of our key ingredients.