The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\mathbb{R}^{3}$

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $x_1,\ldots,x_n$ be distinct points of $\mathbb R^3$, with $n\geq2$. For $i\ne j$, put $v_{ij}=(x_j-x_i)/\|x_j-x_i\|\in S^2$. Fix a stereographic identification $S^2\cong\mathbb{CP}^1$ and write the image of $v_{ij}$ as $[a_{ij}:b_{ij}]$. Choose any nonzero homogeneous linear form $\ell_{ij}(U,V)=b_{ij}U-a_{ij}V$ vanishing at that projective point. Define the homogeneous polynomial $p_i(U,V)=\prod_{j\ne i}\ell_{ij}(U,V)$ of degree $n-1$. Its coefficients in the basis $U^{n-1},U^{n-2}V,\ldots,V^{n-1}$ form the $i$th column of an $n\times n$ complex matrix $M$. Rescaling the chosen linear forms rescales columns by nonzero factors and therefore does not affect invertibility. Homogeneous coordinates handle directions mapping to infinity.

Statement recorded in the supplied source.

Is $\det M\ne0$ for every $n\geq2$ and every configuration of distinct points $x_1,\ldots,x_n\in\mathbb R^3$? Equivalently, are $p_1,\ldots,p_n$ linearly independent? [S2]

Short English statement

Do the direction polynomials associated with any distinct points in three-dimensional space always form an invertible coefficient matrix?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200059 (AMR-071-0059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michael Atiyah and Paul Sutcliffe, The Geometry of Point Particles (2001). Section 2, Conjecture 1, and Section 3, pp. 3–5. https://arxiv.org/pdf/hep-th/0105179v2.