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Denoting by $C_n(X)$ the configuration space of $n$ distinct points in $X$,with $X$ being either Euclidean $3$-space $\mathbb{E}^3$ or hyperbolic$3$-space $\mathbb{H}^3$ or $\mathbb{C}P^1$ , by $\mathscr{P}_{k,d}$ the vectorspace of homogeneous complex polynomials in the variables $z_0, \ldots, z_k$ ofdegree $d$, and by $\mathrm{Obs}^n_d$ the set of all $d$-subsets of$\{1,\ldots,n\}$, the symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^3)$ bypermuting the $n$ points and also acts in a natural way on $\mathrm{Obs}^n_d$.With $n = k+d$, the space $\mathscr{P}_{k,d}$ has dimension $\binom{n}{d}$,which is also the number of elements in $\mathrm{Obs}^n_d$. It is thus naturalto ask the following question. Is there a family of continuous maps $f_I:C_n(X) \to \mathbb{P}\mathscr{P}_{k,d}$, for $I \in \mathrm{Obs}^n_d$ (here$\mathbb{P}$ is complex projectivization), which satisfies$f_I(\sigma.\mathbf{x}) = f_{\sigma.I}(\mathbf{x})$, for all $\sigma \in\Sigma_n$ and all $\mathbf{x} \in C_n(X)$, and such that, for each $\mathbf{x}\in C_n(X)$, the polynomials $f_I(\mathbf{x})$, for $I\in \mathrm{Obs}^n_d$,each defined up to a scalar factor, are linearly independent over $\mathbb{C}$?We provide two closely related smooth candidates for such maps for each of thetwo cases, Euclidean and hyperbolic, which would be solutions to the aboveproblem provided a linear independence conjecture holds. Our maps are naturalextensions of the Atiyah-Sutcliffe maps. Moreover, we get two constructions ofactual solutions of the above problem for $X = \mathbb{C}P^1$, as we provelinear independence for these last two constructions. These last twoconstructions are classical in character, and can be viewed as higherdimensional versions of Lagrange polynomial interpolation. They appear to benew.

Configuration spaces of points, symmetric groups and polynomials of several variables