Verification of Atiyah's Conjecture for some Nonplanar Configurations with Dihedral Symmetry

Dragomir_Ž. Đoković

To an ordered $N$-tuple of distinct points in the three-dimensional Euclidean space, Atiyah has associated an ordered $N$-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 $L$.