Ranks and symmetric ranks of cubic surfaces

Abstract

We study cubic surfaces as symmetric tensors of format 4 × 4 × 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.

Decomposability of Tensors

“…The question raised by Comon asks if whether such an inequality is actually an equality. Affirmative answers were given in several cases (see [176][177][178][179][180]). In [181], Shitov found an example (a cubic in 800 variables) where the inequality (29) is strict.…”

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

“…The question raised by Comon asks if whether such an inequality is actually an equality. Affirmative answers were given in several cases (see [179,176,180,178,177]). In [181], Shitov found an example (a cubic in 800 variables) where the inequality (5.4) is strict.…”

Lower bounds on the rank and symmetric rank of real tensors

“…To date, these large tensors are the only known counterexamples. In comparison, the agreement of rank and symmetric rank was shown for small tensors in [Sei19,Sei20]. The problem of finding a minimal size, or minimal rank, counterexample to Comon's conjeture remains unsolved.…”