Ranks and symmetric ranks of cubic surfaces. (November 2020)

• Record Type: Journal Article
• Authors: Seigal, Anna
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 304
• Page End: 317
• Publication Date: 2020-11
• Subjects: 14Q10 -- 15A69 -- 51N35; Tensor rank -- Cubic surface; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.10.001
• Languages: English
• ISSNs: 0747-7171

Abstract

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.