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 $G= { \rm GL}(d,\mathbb{ K})$.
Categories: math.CA math.GR