Characterization of polynomials by their invariance properties
J. M. Amira, Ya-Qing Hu
TL;DR
The paper develops a general framework to characterize ordinary polynomials on $\mathbb{R}^d$ (and $\mathbb{K}^d$ for characteristic-0 fields) as elements of finite-dimensional spaces of functions that are invariant under translations and under actions of a subgroup $G\subset GL(d)$. A central tool is the condition that the Minkowski difference $\Lambda_{G,z_0}=Gz_0-Gz_0$ has nonempty interior, which forces all invariant functions in the space to be ordinary polynomials; the approach is then specialized to classical groups, including $\mathbf{O}(p,q)$ and $\mathrm{Sp}(2d,\mathbb{R})$, via dimension-reduction and separability arguments. The results extend to distributions with appropriate adaptation (via Anselone--Korevaar) and yield polynomial characterizations through invariant finite-dimensional subspaces, providing both structural insights and quantitative bounds on functional degree. The work illuminates how invariance under broad linear symmetries rigidly constrains function spaces to polynomials, with implications for understanding classical group actions in real and distributional settings.
Abstract
We prove that certain classical groups $G\subseteq {\rm GL}(d,\mathbb{R}^d)$ serve to characterize ordinary polynomials in $d$ real variables as elements of finite-dimensional subspaces of $C(\mathbb{R}^d)$ that are invariant by changes of variables induced by translations and elements of $G$. We also show that, if the field $\mathbb{K}$ has characteristic $0$, the elements of $\mathbb{K}[x_1,\cdots,x_d]$ admit a similar characterization for $G= {\rm GL}(d,\mathbb{K})$.