The Waring Rank of the 3 x 3 Permanent

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree d (with coefficients in an algebraically closed field of characteristic zero) is known only for d ≤ 4, and the best upper bound that is known for d = 5 is 12. In this work, we lower the upper bound to 10 (and there are evidences that it is likely sharp).

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Rank and border rank of Kronecker powers of tensors and Strassen's laser method

Computational Complexity ◽

10.1007/s00037-021-00217-y ◽

2021 ◽

Vol 31 (1) ◽

Author(s):

Austin Conner ◽

Fulvio Gesmundo ◽

Joseph M. Landsberg ◽

Emanuele Ventura

Keyword(s):

Matrix Multiplication ◽

Upper Bounds ◽

Positive Direction ◽

Laser Method ◽

Waring Rank ◽

Border Rank ◽

Symmetric Version

AbstractWe prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor $$T_{cw,q}$$ T c w , q is the square of its border rank for $$q > 2$$ q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for $$q > 4$$ q > 4 . This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, $$T_{skewcw,q}$$ T s k e w c w , q . For $$q = 2$$ q = 2 , the Kronecker square of this tensor coincides with the $$3\times 3$$ 3 × 3 determinant polynomial, $$\det_{3} \in \mathbb{C}^{9} \otimes \mathbb{C}^{9} \otimes \mathbb{C}^{9}$$ det 3 ∈ C 9 ⊗ C 9 ⊗ C 9 , regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $$\det_3$$ det 3 , exhibiting a strict submultiplicative behaviour for $$T_{skewcw,2}$$ T s k e w c w , 2 which is promising for the laser method.We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $$\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}$$ C 3 ⊗ C 3 ⊗ C 3 .

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A bound for the Waring rank of the determinant via syzygies

Linear Algebra and its Applications ◽

10.1016/j.laa.2019.11.007 ◽

2020 ◽

Vol 587 ◽

pp. 195-214

Author(s):

Mats Boij ◽

Zach Teitler

Keyword(s):

Waring Rank

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Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

Mediterranean Journal of Mathematics ◽

10.1007/s00009-020-01609-0 ◽

2020 ◽

Vol 17 (6) ◽

Author(s):

Alexandru Dimca ◽

Gabriel Sticlaru

Keyword(s):

Symmetric Tensors ◽

Waring Rank

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The Hessian Map

International Mathematics Research Notices ◽

10.1093/imrn/rnaa288 ◽

2020 ◽

Author(s):

Ciro Ciliberto ◽

Giorgio Ottaviani

Keyword(s):

Maximal Rank ◽

General Properties ◽

Waring Rank

Abstract In this paper, we study the Hessian map $h_{d,r}$, which associates to any hypersurface of degree $d$ in ${{\mathbb{P}}}^r$ its Hessian hypersurface, and the general properties of this map and prove that $h_{d,1}$ is birational onto its image if $d\geqslant 5.$ We also study in detail the maps $h_{3,1}$, $h_{4,1}$, and $h_{3,2}$ and the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, proving that this restriction is injective as soon as $r\geqslant 2$ and $d\geqslant 3$, which implies that $h_{3,3}$ is birational onto its image. We also prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, as soon as $r\geqslant 2$ and $d\geqslant 3$.

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