Gaps in the pairs (border rank,symmetric rank) for symmetric tensors
Edoardo Ballico
固定整数$m \ge 2$、$s\ge 5$和$d\ge 2s+2$。在这里，我们描述了可能的对称张量秩$\le 2d+s-7$的所有对称张量（或齐次$d$多项式）在$m+1$变量与边界秩$s$。
Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$.
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Rank and border rank of real ternary cubics
本文给出了直到SL（3）-作用为止的实三元三次型的秩和边秩分类，并研究了它们与复数形式的区别。
On tensors of border rank $l$ in $\C^{m\times n\times l}$
我们研究边界秩为$l$的$\C^{m\times n\times l}$中的张量。我们刻画了边界秩最多为4的$\C^{3\times 3\times 4}$和$\C^{4\times 4\times 4}$中的张量。
We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound extends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.
On differences between the border rank and the smoothable rank of a polynomial
我们考虑比Veronese品种更高的正割品种。第r个正割簇上的大多数点都由Veronese簇中包含的长度为r的有限概型表示-实际上，对于泛点，它只是r个不同点的并集。现代的说法是：对于大多数多项式，可平滑秩等于边界秩。这个性质对于研究割线簇非常有用，特别是当可平滑秩等于所讨论的割线簇的所有点的边界秩时。在本文中，我们研究那些特殊点的光滑秩不等于边界秩。特别是，我们展示了一个显式的例子，一个三次在五个变量与边界秩5和光滑秩6。我们还证明了所有的三次最多四个变量的光滑秩等于边界秩。
On the ranks and border ranks of symmetric tensors
受信号处理、计算复杂性和其他领域中出现的问题的启发，我们使用几何方法研究了对称张量的秩和边界秩。我们为对称张量的秩提供了改进的下界（即，齐次多项式）通过考虑由多项式定义的超曲面的奇异性而获得。我们获得正常形式的多项式的边界秩高达5，并计算或绑定的几类多项式，包括单项式，行列式和永久的秩。
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, …
Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border …
Border rank bounds for $GL(V)$-invariant tensors arising from matrices of constant rank
本文研究由常秩矩阵空间产生的 $GL(V)$-不变张量，利用 Young flattenings 证明新的边界秩下界，并确定三个张量的边界秩。
A comparison of different notions of ranks of symmetric tensors
本文引入了对称张量的秩的各种概念，即：秩、边秩、次秩、广义秩、概型长度、边概型长度、扩张秩和光滑秩。我们分析了这些职级引起的分层。这些等级之间的相互关系，使我们能够描述所有等级之间的等级关系。证明了秩、边秩、延拓秩和catalecticant秩之间的严格不等式是可能的。并且证明了格式长度、广义秩和扩张秩是一致的。
On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank
令$σ_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$，$b(m+1) < \binom{m+d}{m}$，表示$m+1$变量中所有次数$d$实齐次多项式的集合（即格式为$(m+1)\times ... \times (m+1)$，$d$倍的实对称张量），其具有在$\mathbb {C}$上的边界秩$b$。它有一个分区到流形的实维$\le b(m+1)-1$，其中的实秩是常数。$σ_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$的典型秩是与维度$b(m+1)-1$的开放部分相关联的秩。在这里，我们分类所有典型的等级时，$b\le 7$和$d, m$不是太小。对于更大的$(m,d,b)$集，我们证明了$b$和$b+d-2$是两个第一典型秩.在$m=1$（真正的二元多项式）的情况下，我们证明了$d$（最大可能的先验值的真正的秩）是一个典型的秩为每个$b$。
The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables
本文给出了边界秩为$5$且本质上依赖于至少5个变量的所有次$d \ge 9$齐次多项式的对称张量秩分层，推广了前人的工作（A. Bernardi，A. Gimigliano，M. Idà，E.巴利科）在较低的边界职级。对于依赖于至少5个变量的多项式，仅5个秩是可能的：$5$、$d+3$、$2d+1$、$3d-1$、$4d-3$，但是秩$3d-1$和$2d+1$中的每一个在两种几何上不同的情况下实现。这些秩由与多项式相关联的一定程度的零维方案$A$唯一地确定。多项式$f$基本上取决于至少5个变量，当且仅当$A$是线性无关的（在所有情况下，$f$基本上取决于恰好5个变量）。多项式具有秩$4d-3$（resp $3d-1$，resp.分别为$2d+1$分别为$d+3$如果$5$有$1$（分别为PDMATHAMV，分别$3$，分别$4$，分别$5$）连接的组件。假设$d\ge 9$保证每个多项式具有唯一确定的关联方案$A$。在每一种情况下，我们描述的维度的家庭的多项式规定的秩，每个不可约的家庭被确定的程度相关的计划$A$的连接组件。
The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicat…
Border rank of powers of ternary quadratic forms
本文主要研究三元二次型任意次幂的边界秩，通过确定配极理想并利用边界配极技术，证明任意次幂 q_3^s 的边界秩等于其中间卡塔莱克坦矩阵的秩 (s+1)(s+2)/2。
Border rank=rank for Kruskal tensors and a Kruskal's theorem for skew decompositions
该研究证明Kruskal张量的边界秩等于秩，给出交替张量的Kruskal条件以证明斜秩分解唯一性，还证明交替Kruskal张量的边界斜秩等于斜秩并给出最小斜秩分解算法。
A $2n^2-log(n)-1$ lower bound for the border rank of matrix multiplication
设M_n表示n × n矩阵的矩阵乘法张量。利用边界替换法结合Koszul证明了M_n的边界秩下界为2n^2-log（n）-1。
Explicit tensors of border rank at least 2n-1
对于奇数n，我写下边界秩为2n-1的C^n \otimes C^n \otimes C^n中的张量，表明Landsberg-Ottaviani的Young平坦化方程的非平凡性。我还研究了Alexeev等张量的边界秩。例如，示出张量它们的张量T_{2^k}，尽管具有等于2^{k+1}-1的秩，但具有等于2 ^k的边界秩，这是任何简洁张量的最小值。我还研究了格里塞方程。
Koszul-Young Flattenings and Symmetric Border Rank of the Determinant
我们提出了新的下界的对称边界秩的n × n行列式的所有n。对于3 × 3永久物给出了进一步的下限。
It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop…
Symmetrization maps and minimal border rank Comon's conjecture
本文针对边界科蒙猜想，利用边界零化性和幂和的边界簇作为技术工具，证明了(ℂⁿ)⊗ᵈ中边界秩为n的大量简洁张量类的该猜想成立，涵盖所有驯服张量及n≤d+1时的所有张量。
Border apolarity and varieties of sums of powers
本文研究幂和簇的边界簇（$\underline{\mathrm{VSP}}$'s），比较其在 Hilbert 概形中的轨迹，引入边界可辨识性概念并提供判据，最后在张量和齐次多项式背景下确定若干实例的 $\underline{\mathrm{VSP}}$'s。
The structure tensor of $\mathfrak{sl}_n$, denoted $T_{\mathfrak{sl}_n}$, is the tensor arising from the Lie bracket bilinear operation on the set of traceless $n \times n$ matrices over $\mathbb{C}$. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of $\mathfrak{sl}_n$ may provide further insight into the complexity of matrix multiplication and the "hay in a haystack" problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the …
Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor
本文详细证明了小 Coppersmith-Winograd 张量的 Kronecker 平方的边界秩为 (q+2)^2，并指出该方法有望推广到 Kronecker 立方及更高次幂，支持猜想值 (q+2)^m。
On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry
我们建立了矩阵乘法张量和其他具有对称性的张量的边界秩算法的基本信息。我们证明了边界秩算法的对称张量（如矩阵乘法和行列式多项式）来在家庭，包括代表与正常形式。这些范式将有助于开发新的有效算法，并证明较低的复杂性界限。我们推导出一个边界秩版本的替代方法用于证明张量秩的下界。我们使用这种边界替换方法和标准形将矩阵乘法的边界秩的下界提高1，达到2n^2- n+1。我们还指出了困难，这将是巨大的障碍，因为“野生”结构的希尔伯特计划点的张量的复杂性较低的界限，未来的进展。
We write down an explicit sequence of tensors in $C^m\otimes C^m\otimes C^m$, for all $m$ sufficiently large, having border rank at least $2.02m$, overcoming a longstanding barrier. We obtain our lower bounds via the border substitution method.
Polynomials of minimal border rank
利用Gorenstein代数的迭代乘法张量与极小可光滑秩齐次多项式之间的对应，分类了至多7个变量中足够高次数的极小边界秩多项式。
Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors
本文第一部分研究了欧几里得空间R^n中向量关于闭半代数集C和给定半代数范数的最佳逼近。假设给定的范数及其对偶范数是可微的，我们证明了超曲面外的最佳逼近是唯一的。然后，我们研究的情况下，C是一个不可约的品种和近似是关于欧几里德规范。我们证明了对于R^n中x中的一般点，x到C的距离函数的临界点的个数在上有界于一个相关优势映射的度.如果C在P（C^n）中诱导出一个光滑射影簇，那么这个度就是V_P上相应向量丛的顶Chern数.然后我们研究了当一个对称张量的最佳k（> 1）-边秩逼近是对称的时的问题.我们证明了在一定的维数条件下，存在一个对称张量的开半代数集，它的最佳k-边秩是唯一的和对称的。
Border rank lower bounds beyond weak border apolarity
本文首次展示超越弱边界对偶性的边界秩下界示例，证明某些张量的理想列表虽非空但无一满足全部条件，涵盖矩阵乘法相关张量及群论相关案例。
The border Waring rank of $x_1\cdots x_n$ is $2^{n-1}$
本文证明特征零域上单项式 $x_1\cdots x_n$ 的边界 Waring 秩为 $2^{n-1}$，表明极化恒等式为最优分解，下界证明借助非对称的行列式张量的高阶 Koszul 展平。
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $ω$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$ permanent, and could potentially be used to show $ω=2$. We prove the negative result for complexity theory that its border rank is $16$, resolving a longstanding problem. Regarding its $q=4$ skew cousin in $ C^5\otimes C^5\otimes C^5$, which could potentially be used to prove $ω\leq 2.11$, we show the border rank of its Kronecker square is at most $42$, a remarkable…
Ranks and Symmetric Ranks of Cubic Surfaces
我们研究三次曲面的格式$4 \times 4 \times 4$的对称张量。本文研究了三次曲面的非对称张量秩和对称Waring秩，证明了这两个概念在复数上是一致的。相应的代数问题涉及边界秩。我们证明了三次曲面的非对称边界秩与对称边界秩是一致的。作为我们的分析的一部分，我们得到极小理想发电机的对称模拟正割品种从鲑鱼猜想。我们还给出了一个测试的对称秩的非零的某些判别式。结果推广到所有大小的三阶张量，当对称秩最多为7时，秩与对称秩相等;当对称边界秩最多为5时，边界秩与对称边界秩相等.我们还通过实替换方法研究了实秩。
We prove that the border rank of the Kronecker square of the little Coppersmith-Winograd tensor $T_{cw,q}$ is the square of its border rank for $q > 2$ and that the border rank of its Kronecker cube is the cube of its border rank for $q > 4$. This answers questions raised implicitly in [Coppersmith-Winograd, 1990] and explicitly in [Bläser, 2013] 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 poten…
Whereas matrix rank is additive under direct sum, in 1981 Schönhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogs of Schönhage's construction for tensors of order four and higher. Schönhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizatio…
New lower bounds for the border rank of matrix multiplication
n × n矩阵的矩阵乘法算子的边界秩是其复杂性的标准度量。利用代数几何和表示论的技巧，我们证明了边界秩至少为2n^2-n。我们的界优于之前的下界（由于Lickteig在1985年）3/2 n^2+ n/2 - 1，对所有n>2。通过寻找小边界秩的双线性映射必须满足的新方程，即，新方程的割线品种的三重Segre产品，矩阵乘法无法满足。
The border rank of the $4 \times 4$ determinant tensor is twelve
该研究通过特定方法证明复数域上4×4行列式张量的边界秩至少为12，结合已知上界确定其边界秩为12，明确了该张量的边界秩精确值。
Equations for lower bounds on border rank
本文给出了确定边界秩至多为r的双线性映射簇的理想中多项式的新方法。我们将这些方法应用于几种情况，包括双线性映射空间C^4 x C^4 -> C^4中r = 6的情况。这个双线性映射空间包括2 × 2矩阵的矩阵乘法算子M_2。我们证明了这些新得到的多项式在矩阵乘法算子M_2上不为零，从而给出了2 × 2矩阵乘法的边秩为7的新证明.其他的例子被认为是与如何实现的方法的解释。
On the geometry of border rank algorithms for n x 2 by 2 x 2 matrix multiplication
研究n×2与2×2矩阵乘法的边界秩算法几何特性，提出新的分析方法和优化策略。
Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the the cp rank, symmetric rank, border rank, symmetric border rank, and Vandermonde rank all coincide with each other. In particular, this implies that the Comon's conjecture is true for generic Hankel tensors when the order is even or three. Some open questions are also posed.
Fixed-parameter debordering of Waring rank
本研究首次获得多项式Waring秩的上界，该上界关于边界Waring秩指数增长、关于次数线性增长，解决了去边界化问题，并改进了常数边界Waring秩情形下的结果。
On degeneration of tensors and algebras
在所有目前的渐近快速算法矩阵乘法的一个重要的构建块是张量与低边界秩，即张量的边界秩是相等或非常接近其大小。为了找到新的渐近快速矩阵乘法算法，理解那些边界秩尽可能小的张量，即所谓的最小边界秩张量，似乎很重要。研究了结合代数的退化与其结构张量在Strassen意义下的退化之间的联系。它使我们能够描述一个开放的子集的$n \times n \times n$张量的最小边界秩的光滑交换代数。我们描述了与Coppersmith-Winograd张量相关的可平滑代数，并证明了Coppersmith和Winograd的“简单构造”中使用的张量的边界秩的下界。
Classification and degenerations of small minimal border rank tensors via modules
本文通过模方法给出了 $m \leq 5$ 时 $1_*$-generic 极小边界秩张量的自包含分类，结合已有结果完成了该范围内所有极小边界秩张量的分类，并完整描述了张量间的退化情况。
Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
本研究证明多体正定与不变张量分解存在秩与边界秩的差距，并利用该差距证明相关概率分布集合非闭集，导致有限时间内无法测试其成员资格，揭示了秩的不稳定性与双体系统的独特性。
Border Ranks of Monomials
本文研究Young展平方法对单项式边界秩的下界极限，给出最优形状，覆盖次数≤6、3变量及变量乘积幂，并指出高次时无法匹配猜想紧上界。
The border rank of the multiplication of two by two matrices is seven
本文证明，近似计算2x2矩阵乘法至少需要七次乘法，即矩阵乘法算子不在三个P^3的Segre乘积的第六割线簇中。
Concise tensors of minimal border rank
借助111-方程与交换矩阵簇结果，引入简洁张量的111-代数不变量，确定了m=5,6时极小边界秩简洁张量的定义方程并完成分类。
Cactus barriers
本文通过仙人掌种群和代数几何方法解释了确定性方法在张量边界秩界限定制中的理论障碍。
Uniformity for limits of tensors
本文研究多线性代数中各类秩的边界秩问题，证明去边界化定理并揭示实现边界秩的极限复杂度不随维数增长，基于GL-簇理论建立统一框架。
Border rank lower bounds for families of GL(V)-invariant tensors
本文针对U⊗S_λV⊗S_μV中GL(V)不变张量族（U为V、Sym²V或∧²V）给出边界秩非平凡下界，解决Wu猜想并推广技术，扩展非最小边界秩例子数量。
Unrestrictions and concise secant varieties
本文基于简洁secant族，探讨了张量理论中border rank与cactus rank的联系，以及与Segre嵌入、Salmon猜想等的关联，提出简洁border apolarity和Sums of Powers varieties的刻画。
On the ranks of the third secant variety of Segre-Veronese embeddings
给出了边秩为3的部分对称张量的秩的上界。在边界秩为3的张量$T\in V_1\otimes \cdots \otimes V_k$的特殊情况下（Segre情况），我们可以证明3和$k-1$之间的所有秩都出现了，如果所有$i$的秩都是$\dim V_i \geq 3$，那么$k$和$2k-1$之间的所有秩也都出现了。
We study three variations of the Waring problem for polynomials, concerning the Waring rank, the border rank and the cactus rank of a form and we show how the Lefschetz properties of the associated algebra affect them. The main tool is the theory of mixed Hessian matrix. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger then the border rank, defined geometrically.
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank…
Topology of tensor ranks
本文研究了由张量秩、边界秩、多线性秩及其对称形式定义的张量集合的路径连通性和同调群，探讨了复数和实数域上的连通性及同调群特性。
Asymptotic rank bounds: a numerical census
本文通过数值方法计算张量的改进渐近秩界，验证了Strassen猜想并揭示了现有数值方法的限制。
We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity ho…
Product Ranks of the $3\times 3$ Determinant and Permanent
证明了$3\times 3$行列式的积秩为$5$，$3\times 3$积积秩为$4$ .作为推论，我们得到了$3 \times 3$行列式和积式的张量秩分别为$5$和$4$.此外，我们表明，边界积秩的$n \times n$永久大于$n$的任何$n\geq 3$。
Border subrank of higher order tensors and algebras
研究高阶结构张量的边界子秩，确定k重矩阵乘法和上三角矩阵乘法的紧界，分析多项式代数、空代数及二次的极化代数的高阶结构张量边界子秩，并证明代数结构张量退化从高阶到低阶的传播。
Computing symmetric rank for symmetric tensors
利用代数几何方法研究了确定对称张量的对称张量秩的问题。我们给算法计算对称秩的$2\times ... \times 2$张量和小边界秩的张量。从几何的角度，我们描述了对称秩层的一些割线品种的Veronese品种。
Symmetric powers: structure, smoothability, and applications
本文研究多项式扭曲幂的边界秩与代数对称幂的可光滑性，证明对称幂可光滑并给出一般情形下边界秩的最优上界，同时引入包络多项式概念并应用于复杂性理论。
Explicit tensors beyond the linear flattening barrier
本文构造了边界秩至少为3n-o(n)的显式张量，并证明线性展平方法无法超越2n-1的下界，从而提供了超越线性展平障碍的显式示例。
Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states
从量子和原子物理学应用的角度，简要回顾了代数几何中张量和多项式分解的工具。应用程序的组装不可区分的两级玻色子原子的例子进行了讨论，使用现代配方的经典西尔维斯特的算法在两个变量的齐次多项式的分解。特别地，我们计算了自旋压缩态的对称秩和对称边界秩，以及它们作为宏观上不同的相干自旋态之和的类薛定谔猫分解; Fock态提供了一个对称秩和对称边界秩不同的态的例子。
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