Is there any known method to approximate a given permutation-equivariant smooth function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ as multivariable polynomial function $p: \mathbb{R}^{n} \to \mathbb{R}^{n}$ on a given domain $D\subset \mathbb{R}^{n}$, let's say, $[-1, 1]^{n}$ or a unit ball centered at the origin? Here permutation-equivariant means that we have $f(\sigma(\mathbf{x})) = \sigma(f(\mathbf{x}))$ for any $\sigma \in S_n$ ($S_n$ acts on $\mathbb{R}^{n}$ by permuting indices), such as max/min function. Minimax approximation means that we want to minimize $L^\infty$ loss in the sense that we want $||p(\mathbf{x}) - f(\mathbf{x})||_{D, \infty}$ to be small. In case of $n=1$ (when permutation equivariant condition is vacuous), Remez's algorithm give a way to find a minimax polynomial approximation on a given interval. It seems that there's no analogue of Remez's algorithm in case of higher dimensions yet, so I wonder if there's any other way to obtain a good approximation. Thanks in advance.

Is there any known method to approximate a given permutation-equivariant smooth function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ as multivariable polynomial function $p: \mathbb{R}^{n} \to \mathbb{R}^{n}$ on a given domain $D\subset \mathbb{R}^{n}$, let's say, $[-1, 1]^{n}$ or a unit ball centered at the origin? Here permutation-equivariant means that we have $f(\sigma(\mathbf{x})) = \sigma(f(\mathbf{x}))$ for any $\sigma \in S_n$ ($S_n$ acts on $\mathbb{R}^{n}$ by permuting indices), such as argmax/min function. Minimax approximation means that we want to minimize $L^\infty$ loss in the sense that we want $||p(\mathbf{x}) - f(\mathbf{x})||_{D, \infty}$ to be small. In case of $n=1$ (when permutation equivariant condition is vacuous), Remez's algorithm give a way to find a minimax polynomial approximation on a given interval. It seems that there's no analogue of Remez's algorithm in case of higher dimensions yet, so I wonder if there's any other way to obtain a good approximation. Thanks in advance.