论阿蒂亚和萨克利夫的著作
On the conjectures of Atiyah and Sutcliffe
摘要：
出于某些问题的物理，阿提亚定义了一个决定性的功能，任何一组$n$不同的点$x_1,..., x_n$在$\mathbb R^3$分配一个复数$D(x_1,..., x_n)$。在一项联合工作中，他和萨克利夫提出了三个关于这个行列式的有趣的命题。他们提供了令人信服的数字证据和行列式的有趣的物理解释。第一个猜想断言行列式永远不会为零，第二个猜想指出它的绝对值至少为1，第三个猜想说$|D(x_1,..., x_n)|^{n-2}\geq \prod_{i=1}^n |D(x_1,..., x_{i-1},x_{i+1},..., x_n)|$。尽管它们的公式很简单，但这些公式似乎是出了名的困难。令$D_n$表示在正则$n-$边形的顶点处求值的Atiyah行列式。我们证明了$\lim_{n\to \infty} \frac{\ln D_n}{n^2}= \frac{7ζ(3)}{2π^2}-\frac{\ln 2}{2}=0.07970479...$，并在这种情况下建立了第二个猜想.进一步证明了凸四边形顶点的第二个猜想和内接四边形顶点的第三个猜想。
英文摘要：
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 $\lim_{n\to \infty} \frac{\ln D_n}{n^2}= \frac{7ζ(3)}{2π^2}-\frac{\ln 2}{2}=0.07970479...$ 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.
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