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As far as I know, determining the set of polynomial equations whose zero locus gives the k-secant variety to the Segre variety $Seg(\mathbb{P}^{n_1} \times ... \times \mathbb{P}^{n_m})$ is an open problem. Since I'm new to this topic, I would like to summarize the state-of-art of this open problem before diving into some research and calculations to get a clear overview of what is done and what remains to be done.

Obviously, I have already done some research on my own, but being a novice I can't quite differentiate, in what has already been done, what is purely theoretical, and what is calculable in practice, either by hand or by computer. Furthermore, I do not hope for someone to give me a full explanation of everything about this topic (even though it would be great :D). I just need someone with a great overview of this topic to provide me with a short summary and a list of references each dealing with some particular cases.

Thanks in advance for your time! :)

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