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

TL;DR

This paper characterizes 1-dimensional semialgebraic images of closed balls and spheres under polynomial, regular, and regulous maps, revealing that all such images can be obtained from simple base maps on circles or intervals.

Contribution

It provides a complete geometric characterization of 1D images of balls and spheres under polynomial, regular, and regulous maps, including a full description of circle images under Laurent polynomials.

Findings

Images of spheres under polynomial, regular, and regulous maps are obtainable from base maps on circles or intervals.

All polynomial maps from higher-dimensional spheres to S^1 are constant.

Characterization of circle images under Laurent polynomials using previous research.

Abstract

We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ under polynomial, regular and regulous maps for some $n\geq1$. 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. We also alternatively prove that all polynomial maps ${\mathbb S}^k\to{\mathbb S}^1$ are constant if $k\geq2$.