On the one dimensional polynomial, regular and regulous images of closed balls and spheres

Abstract: We present a full geometric characterization of the $1$-dimensional (semi-algebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ (of center the origin and radius $1$) or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ (of center the origin and radius $1$) under polynomial, regular and regulous maps for some $n\geq1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$ such that $S$ is the image under such map of either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$. As a byproduct, we provide a full characterization of the images of ${\mathbb S}^1\subset{\mathbb C}\equiv{\mathbb R}^2$ under Laurent polynomials $f\in{\mathbb C}[{\tt z},{\tt z}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst.