Real Rank Geometry of Ternary Forms

TL;DR

This paper investigates the real rank geometry of ternary forms, characterizing the set of sums of powers with generic complex rank, and determining boundaries for various degrees, revealing complex algebraic structures.

Contribution

It provides complete characterizations for quadrics and cubics, and identifies the real rank boundary for quintics, along with partial results for quartics, sextics, and septics.

Findings

Real rank equals the generic complex rank for certain ternary forms.

The real rank boundary for quintics is a degree 168 hypersurface.

Stratification of real varieties of sums of powers via hyperdeterminant-derived discriminants.

Abstract

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we determine the real rank boundary: it is a hypersurface of degree 168. For quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants.