  We compute the equation of the 7-secant variety to the Veronese variety
(P^4,O(3)), its degree is 15. This is the last missing invariant in the
Alexander-Hirschowitz classification.
  It gives the condition to express a homogeneous cubic polynomial in 5
variables as the sum of 7 cubes (Waring problem). The interesting side in the
construction is that it comes from the determinant of a matrix of order 45 with
linear entries, which is a cube. The same technique allows to express the
classical Aronhold invariantof plane cubics as a pfaffian.
