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Verification and Strengthening of the Atiyah-Sutcliffe Conjectures for several Types of Configurations

\begin{; ; ; ; abstract}; ; ; ; In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for $n = 3, 4$ and for general $n$ only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D.{; ; ; ; \DJ}; ; ; ; okovi\'{; ; ; ; c}; ; ; ; ). Interestingly the conjecture C2 (and also stronger C3) is not yet proven even for arbitrary four points in a plane. So far we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra. We have also proposed a strengthening of conjecture C3 for configurations of four points (Four Point Conjectures). For almost collinear configurations (with all but one point on a line) we propose several new conjectures (some for symmetric functions) which imply C2 and C3. By using computations with multi- Schur functions we can do verifications up to $n=9$ of our conjectures. We can also verify stronger conjecture of {; ; ; ; \DJ}; ; ; ; okovi\' c which imply C2 for his nonplanar configurations with dihedral symmetry.

configuration of points ; Atiyah determinant

nije evidentirano