# The Atiyah-Sutcliffe Determinant

## TL;DR

The paper introduces a new general formula for the Atiyah-Sutcliffe determinant function applicable for any integer n ≥ 2, involving higher degree cross-ratios, and proposes a conjecture about its structure.

## Contribution

It provides the first known general formula for the Atiyah-Sutcliffe determinant and conjectures its expression as a rational combination of simple, $SO(3)$-invariant factors.

## Key findings

- New formula for the Atiyah-Sutcliffe determinant for all n ≥ 2
- Conjecture that the determinant is a rational linear combination of two simple invariant factors
- Derivation of a purely angular formula for n=4 as an application

## Abstract

We present a general formula for the Atiyah-Sutcliffe determinant function, which holds for any integer $n \geq 2$, as a global factor times a sum of terms, with each term similar to a higher degree cross-ratio. The formula is to our knowledge new. We also conjecture that the Atiyah-Sutcliffe determinant is a rational linear combination of products of factors of only two simple types, each of them manifestly $SO(3)$-invariant. This allows us to obtain a conjectural purely angular formula for the determinant for $n=4$, as an illustration of how our conjecture can be applied.