Ranks and Symmetric Ranks of Cubic Surfaces

TL;DR

This paper investigates the ranks of cubic surfaces viewed as symmetric tensors, establishing the equivalence of various rank notions over complex numbers and providing algebraic and geometric tools for their analysis.

Contribution

It proves the equality of tensor rank and symmetric rank for cubic surfaces over complex numbers and extends these results to tensors of all sizes with bounded ranks.

Findings

Non-symmetric and symmetric ranks coincide for cubic surfaces.

Non-symmetric border rank equals symmetric border rank for cubic surfaces.

Results extend to tensors of all sizes with bounded symmetric ranks.

Abstract

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method.