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

We present a full geometric characterization of the one-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls ${\overline{B}}_n\subset R^n$ or $n$-dimensional spheres $S^n\subset R^{n+1}$ under polynomial, regular, and regulous maps for some $n⩾1$. In all the previous cases, one can find a new polynomial, regular, or regulous map with domain either ${\overline{B}}_1:=[-1,1]$ or $S^1$ such that $S$ is the image under such map of either ${\overline{B}}_1:=[-1,1]$ or $S^1$. As a by-product, we provide a full characterization of the images of $S^1\subset C\equiv R^2$ under Laurent polynomials $f\in C[z,z^{-1}]$, taking advantage of some previous works of Kovalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps $S^k\to S^1$ are constant if $k⩾2$.