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

If X P n is a projective non-degenerate variety, the X-rank of a point P 2 P n 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 either of symmetric tensors or of homogeneous polynomials. It allows to classify all the possible integers r that can occur in the minimal decomposition of a homogeneous polynomial of X-border rank 4 (i.e. contained in the fourth secant variety) as a linear combination of powers of linear forms.