Gaps in the pairs (border rank,symmetric rank) for symmetric tensors
  Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$. 
Edoardo Ballico
math.AG