Symmetric tensors with applications to Hilbert schemes

Gaps in the pairs (border rank,symmetric rank) for symmetric tensors

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 $s$.

Apolarity, border rank and multigraded Hilbert scheme

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, …

点 Hilbert 概形的自同构与簇的对称幂

Automorphisms of punctual Hilbert schemes and symmetric powers of varieties

本文分类了复光滑射影曲面，回答了Belmans等人的问题并扩展相关工作，简化为对称幂自同构研究，给出高维簇对称幂自同构自然的条件，刻画Kodaira维数≥1的曲面并探讨簇由点Hilbert概形确定的问题。

The symmetric geometric rank of symmetric tensors

Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of the symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of the singular locus of the hypersurface associated to the tensor. We first derive fundamental properties of the symmetric geometric rank. Then, we study the space of symmetric tensors of prescribed symmetric geometric rank, which are spaces of homogeneous polynomials whose corresponding hypersurfaces have a singular locus of bounded codimension.

Separable symmetric tensors and separable anti-symmetric tensors

In this paper, we first introduce the invertibility of even-order tensors and the separable tensors, including separable symmetry tensors and separable anti-symmetry tensors, defined respectively as the sum and the algebraic sum of rank-1 tensors generated by the tensor product of some vectors, say, $v_{1}, v_{2}, \ldots, v_{m}$. We show that the $m!$ sumrands, each in form $v_{σ(1)}\times v_{σ(2)}\times\ldots\times v_{σ(m)}$, are linearly independent if $v_{1},v_{2}, \ldots, v_{m}$ are linearly independent, where $σ$ is any permutation on $\set{1,2,\ldots,m}$. We offer a class of tensors to a…

On the Choi-Lam analogue of Hilbert's 1888 theorem for Symmetric Forms

A famous theorem of Hilbert from 1888 states that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of the form. In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symmetric forms by assuming the existence of psd not sos symmetric $n$-ary quartics for $n \geq 5$. In this paper we complete their proof by constructing explicit psd not sos symmetric $n$-ary quartics for $n \geq 5$.

G-stable rank of symmetric tensors and log canonical threshold

Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the G-stable rank introduced by Derksen is true: the symmetric G-stable rank and G-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the G-stable rank of the defining ideal.

On Waring's problem for partially symmetric tensors

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.

对称理想的对称和等型Hilbert级数

Symmetric and Isotypic Hilbert Series for Symmetric Ideals

本文定义并研究了可数变量多项式环中对称理想的对称和等型Hilbert级数，证明了在温和支撑条件下对称Hilbert级数为有理函数，且该有理性质可推广到每个不可约表示的等型Hilbert级数。

Lower bounds on the rank and symmetric rank of real tensors

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves…

张量的特征多项式与特征值

Characteristic polynomials and eigenvalues of tensors

该研究奠定张量特征多项式研究的几何基础，证明特定维度与阶数的对称张量中共享给定特征多项式的张量数量有限，并针对两类情形下相关张量簇的维数提出精确猜想。

关于某些4生成伪2对称单项式曲线的切锥的Hilbert级数

On the Hilbert Series of the tangent cones for some 4-generated pseudo 2 symmetric monomial curves

本文研究了某些4生成伪对称单项式曲线的非Cohen-Macaulay切锥的Hilbert级数，通过显式计算证明了Hilbert函数的非递减性，并计算了相关环面理想的标准基。

Discrete torsion, symmetric products and the Hilbert scheme

We combine our results on symmetric products and second quantization with our description of discrete torsion in order to explain the ring structure of the cohomology of the Hilbert scheme of points on a K3 surface. This is achieved in terms of an essentially unique symmetric group Frobenius algebra twisted by a specific discrete torsion. This twist is realized in form of a tensor product with a twisted group algebra that is defined by a discrete torsion cocycle. We furthermore show that the form of this cocycle is dictated by the geometry of the Hilbert scheme as a resolution of singularities…

On the symmetric subscheme of Hilbert scheme of points

We consider the Hilbert scheme Hilb^{d+1}(C^d) of (d+1) points in affine d-space C^d (d > 2), which includes the square of any maximal ideal. We describe equations for the most symmetric affine open subscheme of Hilb^{d+1}(C^d), in terms of Schur modules. In addition we prove that Hilb^{d+1}(C^d) is reducible for n>d>11.

Computing symmetric rank for symmetric tensors

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

On the partially symmetric rank of tensor products of W-states and other symmetric tensors

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…

一类循环对称张量的正定性

Positive definiteness of a class of cyclic symmetric tensors

本文研究一类循环对称张量的正定性，建立了四阶三维循环对称张量半正定及四阶n维对称张量正定的充要条件，并据此给出循环与非循环情形的正定性判定，进而得到三元四次齐次多项式的若干不等式。

Moduli of linear representations, symmetric products and the non commutative Hilbert scheme

In this survey paper we study the relationships between the coarse moduli space which parameterizes the finite dimensional linear representations of an associative alegebra, the non commutative hilbert scheme and the affine scheme which is the spectrum of the abelianization of algebra of the divided powers. In particular we will show a map which specialize to the Hlibert - Chow morphism when the associative algebra is commutative. The extension to the positive characteristic case of some of the results due to L. Le Bruyn on noncommutative desingularization is outlined. The possibility to use o…

对称张量与对称张量秩

Symmetric tensors and symmetric tensor rank

本文研究对称张量的性质，特别是其分解为对称外积向量和的形式，并探讨秩与对称秩的关系以及一般对称秩的已知结果。

On minimal decompositions of low rank symmetric tensors

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2.

Symmetric Tensor Decompositions On Varieties

This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary $X$. As a specific application, Vand…

对称张量的鲁棒特征向量

Robust Eigenvectors of Symmetric Tensors

本文证明对称张量中作为对称分解生成元的特征向量在阶数足够高时是鲁棒的，并引入等角紧框架可分解与等角集可分解张量，表明其可用张量幂方法分解。

对称理想与不变Hilbert概形

Symmetric Ideals and Invariant Hilbert Schemes

本文对零维对称理想进行全局几何研究，通过不变Hilbert概形$\mathrm{Hilb}_\rho^{S_n}(\mathbb{C}^n)$，在置换模及低维表示情形下证明其不可约性或光滑性，并分类齐次对称理想，核心工具为更高Specht多项式。

Hilbert schemes and symmetric products: a dictionary

Given a closed complex manifold $X$ of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings $\orbsym$ of the symmetric products. We present constructions and establish results on the rings $\orbsym$ including two sets of ring generators, universality and stability, as well as connections with vertex operators and W algebras. These are independent of but parallel to the main results on the cohomology rings of the Hilbert schemes of points on surfaces as developed in our earlier works joint with W.-P. Li. We introduce a deformation o…

生成多项式与对称张量分解

Generating Polynomials and Symmetric Tensor Decompositions

本文研究对称张量分解，通过生成多项式描述其递归关系，并利用生成矩阵计算分解，探讨其性质及高效算法。

Orthogonal Decomposition of Symmetric Tensors

A real symmetric tensor is orthogonally decomposable (or odeco) if it can be written as a linear combination of symmetric powers of $n$ vectors which form an orthonormal basis of $\mathbb R^n$. Motivated by the spectral theorem for real symmetric matrices, we study the properties of odeco tensors. We give a formula for all of the eigenvectors of an odeco tensor. Moreover, we formulate a set of polynomial equations that vanish on the odeco variety and we conjecture that these polynomials generate its prime ideal. We prove this conjecture in some cases and give strong evidence for its overall co…