Geometric lower bounds for generalized ranks

We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of [LT10] and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by the “non-Abelian” catalecticants recently introduced by Landsberg and Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous polynomials (partially symmetric tensors); a special case is the simultaneous Waring decomposition problem for a linear system of polynomials. We generalize the classical Apolarity Lemma to multihomogeneous polynomials and give some more general statements. Finally we revisit the lower bound of [RS11], and again generalize it to multihomogeneous polynomials and some more general settings.