Multigraded Apolarity

Gałązka, Maciej

doi:10.48550/arxiv.1601.06211

Cited by 10 publications

(15 citation statements)

References 17 publications

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“…Another apolarity theory which is worth noting is the multigraded apolarity, see [5]. We remark also [10] in which it is showed an apolarity theory for toric varieties that features also the use of Cox rings.…”

Section: Introductionmentioning

confidence: 79%

Schur apolarity

Staffolani¹

2021

Preprint

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Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group SL(V ). This allows to describe decompositions of structured tensors whose elementary elements are tensors that represent flags of the vector space V . The main result is the Schur apolarity lemma which is the analogous of the apolarity lemma of symmetric apolarity theory. Eventually we study the rank tensors of low border rank related to specific varieties giving rise also to simple algorithms.

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Section: Introductionmentioning

confidence: 79%

Schur apolarity

Staffolani¹

2021

Preprint

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“…The latter is done by proving the non-existence of ideals satisfying given properties. Some of the many examples of applying Apolarity Lemmas in both directions are [17,Thm. 1.5], [4, §4], as well as Theorems 1.2, 1.3.…”

Section: Ranks and Apolarity Lemmasmentioning

confidence: 99%

Distinguishing secant from cactus varieties

Gałązka¹,

Mańdziuk²,

Rupniewski³

2020

Preprint

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In this paper we study homogeneous polynomials and vector subspaces of a polynomial ring that are divisible by a large power of a linear form. We compute their cactus and border cactus ranks. We show that for d ≥ 5, the component of the cactus variety κ14(ν d (P 6 )) other than the secant variety σ14(ν d (P 6 )) consists of degree d polynomials divisible by (d − 3)-th power of a linear form. For d ≥ 6 we present an algorithm for deciding whether a point in the cactus variety κ14(ν d (P 6 )) belongs to the secant variety σ14(ν d (P 6 )). Analogously, we show that for d ≥ 5, the component of the Grassmann cactus variety κ8,3(ν d (P 4 )) other than the Grassmann secant variety σ8,3(ν d (P 4 )) consists of subspaces divisible by (d − 2)-th power of a linear form. Finally, for d ≥ 5 we present an algorithm for deciding whether a point in the Grassmann cactus variety κ8,3(ν d (P 4 )) belongs to the Grassmann secant variety σ8,3(ν d (P 4 )).

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“…We briefly present the subject and we refer the reader to [IK99, Ger96, CGO14, BCC + 18] for a complete explanation of this material in the homogeneous setting and to [Tei14,Gał16,GRV16] for the multigraded version.…”

Section: Apolarity Theorymentioning

confidence: 99%

“…The apolarity action and the notion of apolar ideal extend to the multigraded setting and to tensor products (and even in more general settings, see [Gał16]). The polynomial ring Sym…”

Section: Apolarity Theorymentioning

confidence: 99%

Border Waring Rank via Asymptotic Rank

Christandl,

Gesmundo,

Oneto

2019

Preprint

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Multigraded Apolarity

Cited by 10 publications

References 17 publications

Schur apolarity

Schur apolarity

Distinguishing secant from cactus varieties

Border Waring Rank via Asymptotic Rank

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