Completely Reducible Ternary Cubic Forms

TL;DR

This paper establishes a precise criterion for the complete reducibility of ternary cubic forms, showing it occurs if and only if the form's Hessian is proportional to the form itself, thus resolving a 19th-century claim.

Contribution

It proves the necessary and sufficient condition for complete reducibility of ternary cubic forms based on their Hessian, confirming a historical conjecture.

Findings

Complete reducibility characterized by Hessian proportionality

Proof of 19th-century claim on ternary cubic forms

Provides necessary and sufficient conditions for reducibility

Abstract

We discuss various necessary and sufficient conditions for the complete reducibility of a ternary cubic form. In doing so, we prove the claim made in the 19th century that such a form is completely reducible if and only if its Hessian is a multiple of itself.