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

We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls ${\bar{\mathcal{B}}}_n\subset {\mathbb{R}}^n$ or $n$-dimensional spheres ${\mathbb{S}}^n\subset {\mathbb{R}}^{n+1}$ under polynomial, regular and regulous maps for some $n\ge 1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either ${\bar{\mathcal{B}}}_1:=[-1,1]$ or ${\mathbb{S}}^1$ such that $S$ is the image under such map of either ${\bar{\mathcal{B}}}_1:=[-1,1]$ 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 $f\in \mathbb{C}[\mathtt{z},{\mathtt{z}}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps ${\mathbb{S}}^k\to {\mathbb{S}}^1$ are constant if $k\ge 2$.