On the sums of two cubes

TL;DR

This paper investigates the solutions to a specific cubic equation involving homogeneous rational functions, revealing a ring structure isomorphic to a complex quadratic integer ring, thus extending classical elliptic curve addition laws.

Contribution

It completely solves a historic cubic equation and uncovers a novel algebraic structure among its solutions, connecting to elliptic curve operations and complex quadratic rings.

Findings

Solution set has a ring structure isomorphic to ^{2 \u03c0 i/3}

Established a connection between solutions and elliptic curve addition laws

Extended classical results from Vite's 1591 investigation

Abstract

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.