Duality of Multiple Root Loci

Lee, Hwangrae; Sturmfels, Bernd

doi:10.48550/arxiv.1508.00202

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“…Given a partition α of d, the general Chow variety, denoted Chow α PV ⊂ PS d V , consists of completely decomposable homogeneous polynomials of degree d with splitting type α, i.e. all polynomials of the form ℓ α 1 1 • • • ℓ αt t (up to scale), where ℓ i are linear forms in V , (see [41,42,6,13,1]). Chow d PV is the Veronese variety ν d PV .…”

Section: Introductionmentioning

confidence: 99%

“…Lee and Sturmfels recently used projective duality to study the Euclidean distance degrees of coincident root loci [41], following classical work of Hilbert and more recent work [42]. Briand revitalized the study of Brill and Gordan's classical set-theoretic defining equations for the variety of completely reducible forms [6] and Chapalkatti addressed the problem for general coincident root loci [13].…”

Section: Introductionmentioning

confidence: 99%

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Border Ranks of Monomials

Oeding

2016

Preprint

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Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and their non-vanishing provides lower bounds for border ranks of tensors and in particular polynomials. We study monomial-optimal shapes for Young flattenings, which exhibit the limits of the Young flattening method. In particular, they provide the best possible lower bound for large classes of monomials including all monomials up to degree 6, monomials in 3 variables, and any power of the product of variables. On the other hand, for degree 7 and higher there are monomials for which no Young flattening can give a lower bound that matches the conjecturally tight upper bound of Landsberg and Teitler.

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