Stratification of the fourth secant variety of Veronese variety via the symmetric rank

If X ⊂ Pn is a projective non degenerate variety, the X-rank of a point P ∈ Pn is defined to be the minimum integer r such that P belongs to the span of r points of X. We describe the complete stratification of the fourth secant variety of any Veronese variety X via the X-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers r that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of X-border rank 4 (i.e. contained in the fourth secant variety) as a linear combination of either completely decomposable tensors or powers of linear forms respectively.