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

We give the stratification by the symmetric tensor rank of all degree $d \ge 9$ homogeneous polynomials with border rank $5$ and which depend essentially on at least 5 variables, extending previous works (A. Bernardi, A. Gimigliano, M. Id\`{a}, E. Ballico) on lower border ranks. For the polynomials which depend on at 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 two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme $A$ associated to the polynomial. The polynomial $f$ depends essentially on at least 5 variables if and only if $A$ is linearly independent (in all cases $f$ essentially depends on exactly 5 variables). 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 polynomial has a uniquely determined associated scheme $A$. In each case we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme $A$.