关于四点构型的 Atiyah 猜想的新证明
A new proof of Atiyah's conjecture on configurations of four points
AI总结：
本文利用 2-旋量演算与 Hermite 半正定矩阵理论，证明四点构型对应四个多项式的 Gram 矩阵恒正定，从而给出 Atiyah 四点线性无关猜想的新证明。
摘要：
在 2002 年出版的《Surveys in Differential Geometry》第 7 卷以及 2001 年出版的《Philosophical Transactions of the Royal Society A》第 359 卷中，Michael Atiyah 爵士提出了所谓的点构型 Atiyah 问题；该问题可简要描述为如下猜想：与 $\mathbb{R}^3$ 中 $n$ 个不同点的构型在几何上相关联的 $n$ 个多项式（每个多项式的定义至多相差一个相位因子）总是线性无关的。第一个“困难”情形是 $n=4$ 个点，Eastwood 和 Norbury 于 2001 年在《Geometry & Topology》第 2 卷中证明了这一线性无关猜想。我们给出关于四点构型的 Atiyah 线性无关猜想（即 Eastwood 和 Norbury 定理）的一个新证明。我们的证明在于表明：与欧几里得 3 维空间中 4 个点的构型相关联的 4 个多项式的 Gram 矩阵总是正定的。该证明运用了 2-旋量演算以及 Hermite 半正定矩阵理论。
英文摘要：
In Surveys in Differential Geometry, Volume 7, published in 2002 and Philosophical Transactions of the Royal Society A, Volume 359, published in 2001, Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of points, which can be briefly described as the conjecture that the $n$ polynomials (each defined up to a phase factor) associated geometrically to a configuration of $n$ distinct points in $\mathbb{R}^3$ are always linearly independent. The first ``hard'' case is for $n = 4$ points, for which the linear independence conjecture was proved by Eastwood and Norbury in Geometry & Topology (2), in 2001. We present a new proof of Atiyah's linear independence conjecture on configurations of four points, i.e. of Eastwood and Norbury's theorem. Our proof consists in showing that the Gram matrix of the $4$ polynomials associated to a configuration of $4$ points in Euclidean $3$-space is always positive definite. It makes use of $2$-spinor calculus and the theory of hermitian positive semidefinite matrices.
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