Verification of Atiyah's conjecture for some nonplanar configurations with dihedral symmetry

To an ordered TV-tuple of distinct points in the three-dimensional Euclidean space, Atiyah has associated an ordered TV-tuple of complex homogeneous polynomials in two variables of degree N - 1, each determined only up to a scalar factor. He has conjectured that these polynomials are linearly independent. In this note it is shown that Atiyah's conjecture is true if m of the points are on a line L and the remaining n = N - m points are the vertices of a regular n-gon whose plane is perpendicular to L and whose centroid lies on L.