Atiyah's linear independence conjecture for configurations of points in three-space

Let $(x_1,\ldots,x_N)$ be an ordered $N$-tuple of distinct points in ${\mathbb R}^3$. For each $i\ne j$, stereographic projection of the direction $(x_j-x_i)/|x_j-x_i|$ to ${\mathbb {CP}}^1$ gives a nonzero linear form $l_{ij}=u_{ij}x+v_{ij}y\in{\mathbb C}[x,y]$. Define

$$p_i=\prod_{j\ne i}l_{ij}(x,y),\quad i=1,\ldots,N.$$

Atiyah's conjecture. The polynomials $p_1,\ldots,p_N$ are linearly independent. Equivalently, their $N\times N$ coefficient matrix is nonsingular. Atiyah's conjecture is known for collinear configurations, for $N=3$, for arbitrary four points, and for the nonplanar dihedrally symmetric family treated in this paper; it remains open in general.