On function compositions that are polynomials

TL;DR

This paper studies when polynomials can be expressed as compositions of a polynomial map and an arbitrary function, showing that under certain conditions, the outer function must also be polynomial.

Contribution

It proves that for surjective polynomial maps over algebraically closed fields of characteristic zero, the composed function must be polynomial.

Findings

If the polynomial map is surjective over an algebraically closed field of characteristic zero, then the composed function is necessarily polynomial.

The result characterizes the structure of functions that can be written as compositions with polynomial maps.

Provides conditions under which the outer function in a composition must be polynomial.

Abstract

For a polynomial map $\mathbf{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \mathbf{f}$, where $h: k^m \to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic $0$ and $\mathbf{f}$ is surjective, we will show that $g = h \circ \mathbf{f}$ implies that $h$ is a polynomial.