On the conjectures of Atiyah and Sutcliffe

Details On the conjectures of Atiyah and Sutcliffe On the conjectures of Atiyah and Sutcliffe

2011-02-23

arxiv.org/abs/1102.4662v2

Mathematics

Algebraic Geometry

18 pages, 1 figure; a typo on p. 6 has been corrected and in 2 places some brief explanations have been inserted

Scientific paper

Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and Sutcliffe stated three intriguing conjectures about this determinant. They provided compelling numerical evidence for the conjectures and an interesting physical interpretation of the determinant. The first conjecture asserts that the determinant never vanishes, the second states that its absolute value is at least one, and the third says that $|D(x_1,..., x_n)|^{n-2}\geq \prod_{i=1}^n |D(x_1,..., x_{i-1},x_{i+1},..., x_n)|$. Despite their simple formulation, these conjectures appear to be notoriously difficult. Let $D_n$ denote the Atiyah determinant evaluated at the vertices of a regular $n-$gon. We prove that $\lim_{n\to \infty} \frac{\ln D_n}{n^2}= \frac{7\zeta(3)}{2\pi^2}-\frac{\ln 2}{2}=0.07970479...$ and establish the second conjecture in this case. Furthermore, we prove the second conjecture for vertices of a convex quadrilateral and the third conjecture for vertices of an inscribed quadrilateral.

Affiliated with

Also associated with

No associations

Rating

On the conjectures of Atiyah and Sutcliffe does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the conjectures of Atiyah and Sutcliffe, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the conjectures of Atiyah and Sutcliffe will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-174264