On function compositions that are polynomials

Abstract

For a polynomial map $\mathbf{f}:k^n\to k^m$ ($k$ a field), we investigate those polynomials $g\in k[t_1,\dots ,t_n]$ 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.