Proof of Atiyah's conjecture for two special types of configurations

Abstract

To an ordered N-tuple (x1, . . . , xN ) of distinct points in the three-dimensional Euclidean space Atiyah has associated an ordered N-tuple of complex homogeneous polynomials (p1, . . . , pN ) in two variables x, y of degree N − 1, each pi 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 for two special configurations of N points. For one of these configurations, it is shown that a stronger conjecture of Atiyah and Sutcliffe is also valid.