Harmonic exponential terms are polynomial

Abstract Let n be a positive integer and f belong to the smallest ring of functions $\mathbb R^n\to \mathbb R$ that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists $d\in \mathbb N$ such that for all $m\in \{0,\dots , n\}$ and $c\in \mathbb R^{m}$ , if $x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ is harmonic, then it is polynomial of degree at most d . In particular, f is polynomial if it is harmonic.