Geometric lower bounds for generalized ranks

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.

43 pages. v2: minor changes

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• The asymptotic leading term for maximum rank of ternary forms of a given degree
• A note on the cactus rank for Segre-Veronese varieties
• Symmetric tensors: rank and Strassen's conjecture
• Symmetric tensors: rank, Strassen's conjecture and e-computability
• Tensor decomposition methods for correlated electron pairs
• Border Waring Rank via Asymptotic Rank
• Identifying limits of ideals of points in the case of projective space
• $e$-computable forms and the Strassen conjecture
• Semialgebraic Geometry of Nonnegative Tensor Rank