Rank and border rank of real ternary cubics

TL;DR

This paper classifies real ternary cubic forms based on their rank and border rank, comparing these classifications with the complex case to highlight differences.

Contribution

It provides a comprehensive classification of real ternary cubics by rank and border rank under SL(3) action, extending known complex results.

Findings

Classification of real ternary cubics by rank and border rank

Differences between real and complex cases identified

Enhanced understanding of the geometric properties of these forms

Abstract

In this work we give the classification of real ternary cubic forms with respect to rank and border rank up to SL(3)-action and we examine the differences with the complex case.