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We give the stratification by the symmetric tensor rank of all degree $d \ge9$ homogeneous polynomials with border rank $5$ and which depend essentially onat least 5 variables, extending previous works (A. Bernardi, A. Gimigliano, M.Id\`{a}, E. Ballico) on lower border ranks. For the polynomials which depend onat least 5 variables only 5 ranks are possible: $5$, $d+3$, $2d+1$, $3d-1$,$4d-3$, but each of the ranks $3d-1$ and $2d+1$ is achieved in twogeometrically different situations. These ranks are uniquely determined by acertain degree 5 zero-dimensional scheme $A$ associated to the polynomial. Thepolynomial $f$ depends essentially on at least 5 variables if and only if $A$is linearly independent (in all cases $f$ essentially depends on exactly 5variables). The polynomial has rank $4d-3$ (resp $3d-1$, resp. $2d+1$, resp.$d+3$, resp. $5$) if $A$ has $1$ (resp. $2$, resp. $3$, resp. $4$, resp. $5$)connected components. The assumption $d\ge 9$ guarantees that each polynomialhas a uniquely determined associated scheme $A$. In each case we describe thedimension of the families of the polynomials with prescribed rank, eachirreducible family being determined by the degrees of the connected componentsof the associated scheme $A$.

The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables