Cayley-Bacharach Formulas

TL;DR

This paper provides explicit rational formulas for the ninth point determined by the Cayley-Bacharach Theorem, which states that all cubic curves through eight points pass through a unique ninth point.

Contribution

It derives explicit rational formulas for the ninth point in the Cayley-Bacharach configuration, making the theorem computationally accessible.

Findings

Explicit rational formulas for the ninth point

Enhanced computational understanding of Cayley-Bacharach configurations

Potential applications in algebraic geometry computations

Abstract

The Cayley-Bacharach Theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.