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

Abstract

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.