Polynomials of small symmetric slice rank

I will explain some results on secant varieties of the variety of reducible polynomials, focusing on the case of cubics. In particular, I will illustrate some upper bound on the smallest possible number of variables required to obtain set theoretic equations for these varieties. The methods are based on the representation theory of certain shifted partial derivatives maps and on the classical geometry of defective varieties in small dimension. This is based on join work with C. Flavi, A. Oneto and E. Ventura.