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

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