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- W4298587797 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." @default.
- W4298587797 created "2022-10-02" @default.
- W4298587797 creator A5010171429 @default.
- W4298587797 date "2018-01-16" @default.
- W4298587797 modified "2023-10-06" @default.
- W4298587797 title "Ranks and Symmetric Ranks of Cubic Surfaces" @default.
- W4298587797 doi "https://doi.org/10.48550/arxiv.1801.05377" @default.
- W4298587797 hasPublicationYear "2018" @default.
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