Geometric lower bounds for generalized ranks

TL;DR

This paper extends geometric lower bounds for polynomial ranks, generalizing previous bounds to broader varieties and multihomogeneous polynomials, with applications to tensor decompositions and polynomial rank estimation.

Contribution

It introduces a generalized geometric lower bound for ranks with respect to arbitrary varieties, extending prior bounds and applying to multihomogeneous polynomials and tensor decompositions.

Findings

Generalized lower bounds for polynomial ranks with respect to arbitrary varieties.

Extended the Apolarity Lemma to multihomogeneous polynomials.

Revised and generalized the Ranestad-Schreyer lower bound for multihomogeneous cases.

Abstract

We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of Landsberg and Teitler 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 Ranestad and Schreyer, and again generalize it to multihomogeneous polynomials and some more general settings.