Characterization of polynomials by their invariance properties

We prove that certain classical groups $G\subseteq \mathrm{G}\mathrm{L}(d,{\mathbb{R}}^d)$ 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 $G=\mathrm{G}\mathrm{L}(d,\mathbb{K})$.