Bounded-rank tensors are defined in bounded degree

Draisma, Jan; Kuttler, Jochen

doi:10.1215/00127094-2405170

Cited by 50 publications

(49 citation statements)

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“…(In fact, in [DK,§7], a conjecture concerning something similar to statements (a)-(c) above is posed.) • Landsberg and Weyman [LW] have obtained results about the ideal of the tangent variety to the Segre, which imply that it is finitely generated as a ∆-ideal.…”

Section: Syzygies Of ∆-Schemesmentioning

confidence: 98%

Syzygies of Segre embeddings and Δ-modules

Snowden¹

2013

Duke Math. J.

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Abstract. We study syzygies of the Segre embedding of P(V1) × · · · × P(Vn), and prove two finiteness results. First, for fixed p but varying n and Vi, there is a finite list of "master p-syzygies" from which all other p-syzygies can be derived by simple substitutions. Second, we define a power series fp with coefficients in something like the Schur algebra, which contains essentially all the information of p-syzygies of Segre embeddings (for all n and Vi), and show that it is a rational function. The list of master p-syzygies and the numerator and denominator of fp can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting n and the Vi vary) certain structure on the space of p-syzygies emerges. We formalize this structure in the concept of a ∆-module. Many of our results on syzygies are specializations of general results on ∆-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.

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Section: Syzygies Of ∆-Schemesmentioning

confidence: 98%

Syzygies of Segre embeddings and Δ-modules

Snowden¹

2013

Duke Math. J.

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“…The set-theoretic theorem above was first proved in [DK13] for kSeg, i.e., for any fixed secant variety of Seg; and a discussion with Snowden led to the insight that our proof generalises to bounded, good, ∆-varieties as in the theorem. Further recent keywords closely related to the topic of this chapter are GL ∞ -algebras [SS12], FI-modules [CEF12], and cactus varieties [BB13].…”

Section: Tensors and ∆-Varietiesmentioning

confidence: 83%

“…This additional source of linear maps along which to pull back equations may allow for an ideal-theoretic version of the theorem. For details see [Sno13,DK13].…”

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confidence: 99%

Noetherianity up to Symmetry

Draisma

2014

Lecture Notes in Mathematics

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Abstract. These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations-whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few "personal favourites": exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

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“…For very general, so-called equivariant models, the fact that on set-theoretic level there exists a bound was proved in [DK09,DK14,DE15]. For the class of G-models that includes all the models introduced in this article, on the level of projective schemes the bounds were obtained in [Mic13].…”

Section: Introductionmentioning

confidence: 99%

Finite phylogenetic complexity ofZpand invariants forZ3
Michałek
¹

2017
European Journal of Combinatorics
2
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Abstract. We study phylogenetic complexity of finite abelian groups -an invariant introduced by Sturmfels and Sullivant [SS05]. The invariant is hard to compute -so far it was only known for Z 2 , in which case it equals 2 [SS05, CP07]. We prove that phylogenetic complexity of any group Z p , where p is prime, is finite. We also show, as conjectured by Sturmfels and Sullivant, that the phylogenetic complexity of Z 3 equals 3.

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Bounded-rank tensors are defined in bounded degree

Cited by 50 publications

References 30 publications

Syzygies of Segre embeddings and Δ-modules

Syzygies of Segre embeddings and Δ-modules

Noetherianity up to Symmetry

Finite phylogenetic complexity ofZpand invariants forZ3
Michałek
¹

2017
European Journal of Combinatorics
2
0
1
0
View full text Add to dashboard Cite

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Bounded-rank tensors are defined in bounded degree

Cited by 50 publications

References 30 publications

Syzygies of Segre embeddings and Δ-modules

Syzygies of Segre embeddings and Δ-modules

Noetherianity up to Symmetry

Finite phylogenetic complexity ofZpand invariants forZ3Michałek1 2017European Journal of Combinatorics2010View full textAdd to dashboardCite

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Finite phylogenetic complexity ofZpand invariants forZ3
Michałek
¹

2017
European Journal of Combinatorics
2
0
1
0
View full text Add to dashboard Cite