We prove that certain classical groups \(G\subseteq {{\,\textrm{GL}\,}}(d,\mathbb {R}^d)\) 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 \(\mathbb {K}[x_1,\dots ,x_d]\) admit a similar characterization for \(G={{\,\textrm{GL}\,}}(d,\mathbb {K})\) .
