Moment-like maps and real algebraic functions with prescribed preimages

We discuss a problem on singularity theory of differentiable (smooth) or real algebraic maps which is different from knowing existence and has been difficult: constructing explcit real algebraic functions. We discuss construction of real algebraic functions with exactly one singular value, the singular points being of definite type, and prescribed preimages of single points. We discuss generalizations of the canonical projection of the unit sphere around the pole and the Morse-Bott functions around the boundaries of the images with preimages diffeomorphic to the torus. This has been discussed in the differentiable (smooth) category since the pioneering study of Sharko in 2006, followed by Masumoto-Saeki, Michalak and so on: the author has first considered the cases respecting the topologies of the preimages of the points where these studies had not done this essentially. Related real algebraic studies have been started by the author essentially in 2020's and the studies are developing, mainly due to the author.