Claudio Fontanari
Here we address a Waring type problem for partially symmetric tensors, extending previous work by Massimiliano Mella in the totally symmetric case of forms. In particular, we provide an explicit answer in lower dimensional cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Eigenvectors of tensors and algorithms for Waring decomposition
本文探讨了计算多项式Waring分解的算法，该算法与割线簇方程和张量特征向量相关，并显式实现了三元一般三次多项式的五立方分解。
We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.
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。
Waring's problem, of expressing an integer as the sum of powers, has a very long history going back to the 17th century, and the problem has been studied in many different contexts. In this paper we introduce the notion of a Waring subspace and a Waring identifiable subspace with respect to a projective algebraic variety $\mathcal X$. When $\mathcal X$ is the Veronese variety, these subspaces play a fundamental role in the theory of symmetric tensors and are related to the Waring decomposition and Waring identifiability of symmetric tensors (homogeneous polynomials). We give several constructi…
We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.
Symmetric tensor decomposition on rational varieties
研究节点位于有理簇上的对称张量的Waring分解，给出存在性的显式刻画和高效分解算法，推广Hankel张量到多元情形，并建立有理曲线上求积公式节点数的新上界。
Structures of the Length Seven Power Sum Decompositions of Ternary Quartics
本文探讨确定给定次数和维度下最大对称秩的路径，通过研究三次四次型的长度七Waring分解结构，为归纳过程奠定基础，并考察了可能具有根本重要性的技术工具。
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…
Weddle loci of linear systems of quadrics and the rank of partially symmetric tensors
研究部分对称张量的秩与Weddle领地奇点数目的关系，通过低秩条件推导出奇点下界并证明其与Jacobsthal数的关联。
Partially symmetric tensors and the non-defectivity of secant varieties of products with a projective line as a factor
该研究在多重次数轻微限制下，证明含k-2个射影直线因子的k>2因子Segre-Veronese簇的所有割线簇具预期维数，等价于对应部分对称张量集的维数计算，结果基于k=2情形及新证的一般射影簇定理。
At the time of writing, the general problem of finding the maximal Waring rank for homogeneous polynomials of fixed degree and number of variables (or, equivalently, the maximal symmetric rank for symmetric tensors of fixed order and in fixed dimension) is still unsolved. To our knowledge, the answer for ternary quartics is not widely known and can only be found among the results of a master's thesis by Johannes Kleppe at the University of Oslo (1999). In the present work we give a (direct) proof that the maximal rank for plane quartics is seven, following the elementary geometric idea of spli…
Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T \otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry become available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information the…
We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing …
A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special…
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. …
Generalized Additive Decompositions of Symmetric Tensors
本文研究对称张量的广义加性分解（GAD），提出衡量其规模的线性代数方法，证明GAD秩与Catalecticant矩阵秩的一致性，给出极零方案的显式描述及最小唯一性条件，开发数值GAD算法并验证其有效性与稳定性。
A new bound on the rank of tensor product of W-states
本文提出关于W态张量积秩的新界，并给出显式的分解方法，改进了现有最佳界。
Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors
该研究针对实对称三阶张量的Comon秩间隙，建立了两个互补的上界，扩展了Friedland的展平秩准则，明确了两个控制函数的取值范围及来源，使界在互补区域有效。
Superfat points and associated tensors
本研究聚焦n维空间中m对称的0维概形及其中的m-超胖点，重点分析n=2时的性质，并以平面2-超胖点为例，探究其在Veronese等簇上定义的簇及对应张量，给出此类概形的首个应用。
We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $\dim V_i \geq 3$ for all $i$'s, then also all the ranks between $k$ and $2k-1$ arise.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。