Sylvester's Algorithm

The Sylvester's algorithm allows to compute the symmetric rank of a symmetric tensor $T\in S^dV$ if $V$ is a two dimensional vector space. In the first part of the talk I will present the versions of this that gives in a finite number of steps the symmetric rank of the tensor without giving an explicit decomposition of it. In the second part of the talk I will show how the same geometric idea can be applied to write effective algorithms also for some higher dimensional cases. What is presented in this talk comes from a work in progress with A. Gimigliano and M. Idà.