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

Dragomir Z. Djokovic
2002 The Electronic Journal of Linear Algebra  
To an ordered N -tuple (x 1 , . . . , x N ) of distinct points in the three-dimensional Euclidean space Atiyah has associated an ordered N -tuple of complex homogeneous polynomials (p 1 , . . . , p N ) in two variables x, y of degree N − 1, each p i 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
more » ... own that a stronger conjecture of Atiyah and Sutcliffe is also valid. Key words. Atiyah's conjecture, Hopf map, Configuration of N points in the three-dimensional Euclidean space, Complex projective line.
doi:10.13001/1081-3810.1081 fatcat:jwtseyuysjgdxbj2ze2kt63bjm