On the ranks of homogeneous polynomials of degree at least 9 and border rank 5


Let f be a degree d \ge 9 homogenous polynomial with border rank 5. We prove that it has rank at most 4d-2 and give better results when f essentially depends on at most 3 variables or there are other conditions on the scheme evincing the cactus and border rank of f. We always assume that f essentially depends on at most 4 variables, because the other case was done by myself in Acta Math. Vietnam. 42 (2017), 509-531.

DOI Code: 10.1285/i15900932v38n2p55

Keywords: symmetric tensor rank; border rank; cactus rank

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