  Comon's conjecture on the equality of the rank and the symmetric rank of a
symmetric tensor, and Strassen's conjecture on the additivity of the rank of
tensors are two of the most challenging and guiding problems in the area of
tensor decomposition. We survey the main known results on these conjectures,
and, under suitable bounds on the rank, we prove them, building on classical
techniques used in the case of symmetric tensors, for mixed tensors. Finally,
we improve the bound for Comon's conjecture given by flattenings by producing
new equations for secant varieties of Veronese and Segre varieties.
