On a problem of Pinkus and Wajnryb regarding density of multivariate polynomials (Q2832819)

We denote by \(\mathbb{R}[x_{1},\ldots, x_{d}]\) the space of all polynomials in \(d\) variables with real coefficients, and by \(\mathbb{Z}_{+}\) the set of all non-negative integers. For \(p \in \mathbb{R}[x_{1},\ldots, x_{d}],\) replacing \(x_{j}\) in \(p\) with the derivative \(D_{x_{j}}\) with respect to the \(j\)-s variable, we associate with \(p\) the differential operator \(p(D) = p(D_{x_{1}}, \ldots, D_{x_{d}}).\) Further, for a fixed \(g \in \mathbb{R}[x_{1},\ldots, x_{d}],\) we set \(\mathcal{P}(g) := \text{span} \{ (g(\cdot +\mathbf{b}))^{k} : \, \mathbf{b} \in \mathbb{R}^{d}, k \in \mathbb{Z}_{+} \}.\)NEWLINENEWLINEThe authors prove the following result, which give an affirmative answer for two former questions posed by A. Pinkus and B. Wajnryb regarding density of certain classes of multivariate polynomials.NEWLINENEWLINE{Theorem.} \, For any \(d\) and any \(g \in \mathbb{R}[x_{1},\ldots, x_{d}]\) the following are equivalent: NEWLINE{\parindent8mmNEWLINE\begin{itemize}\item[(i)] \(\mathcal{P}(g) \neq \mathbb{R}[x_{1},\ldots, x_{d}];\) NEWLINE\item[(ii)] There exists a non-zero polynomial \(q \in \mathbb{R}[x_{1},\ldots, x_{d}]\) such that \(q(D)(g^{k}) = 0\) for all \(k \in \mathbb{Z}_{+};\) NEWLINE\item[(iii)] \(\overline{\mathcal{P}(g)} \neq C(\mathbb{R}^{d})\) (the closure is taken in the topology of uniform convergence on compact subsets of \(\mathbb{R}^{d}\)).NEWLINENEWLINE\end{itemize}}