The rank and symmetric rank of a symmetric tensor may differ.

No abstract

Let S be a set of n × n matrices over a field F. We show that the F-linear span of the words in S of length at most 2n log 2 n + 4n is the full F-algebra generated by S. This improves on the n 2 /3 + 2/3 bound by Paz (1984) and an O(n 3/2 ) bound of Pappacena (1997).Let S be a subset of a finite-dimensional associative algebra A over a field F. An element a ∈ A is said to be a word of length k in S if there are a 1 , . . . , a k ∈ S such that a = a 1 . . . a k . We denote the set of all such words by S k , and we write FS k for the F-linear span of S k . Similarly, FS k will stand for the F-linear span of all the words in S that have length at most k.Definition 1. The length ℓ(S) is the smallest integer k for which FS k is the full subalgebra generated by S. We also define ℓ(A) as the maximum value of ℓ(S), where S runs over all subsets of A that generate A as an F-algebra.

Let A be a matrix with nonnegative real entries. The PSD rank of A is the smallest integer k for which there exist k × k real PSD matrices B 1 , . . . , Bm, C 1 , . . . , Cn satisfying A(i|j) = tr(B i C j ) for all i, j. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.2010 Mathematics Subject Classification. 15A23, 90C22.

Abstract. We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that 6n 7 linear inequalities suffice to describe a convex n-gon up to a linear projection. PreliminariesConsider a convex polytope P ⊂ R n . An extension [5,8] of P is a polytope Q ⊂ R d such that P can be obtained from Q as an image under a linear projection from R d to R n . An extended formulation [8,10] of P is a description of Q by linear equations and linear inequalities (together with the projection). The size [8, 10] of the extended formulation is the number of facets of Q. The extension complexity [8, 10] of a polytope P is the smallest size of any extended formulation of P , that is, the minimal possible number of inequalities in the description of Q. The number of facets of Q can sometimes be significantly smaller [5] than that of P , and this phenomenon can be used to reduce the complexity of linear programming problems useful for numerous applications [3,5,10

The rank and symmetric rank of a symmetric tensor may differ.We work with three-dimensional tensors with complex entries. The entries of such a tensor are indexed by triples (i, j, k) in the Cartesian product of finite indexing sets I, J, K.and symmetric if I = J = K and the value of T (i|j|k) does not change under permutations of (i, j, k). The rank of T is the smallest r for which T can be written as a sum of r simple tensors, and such a representation is called a rank decomposition of T . If T is a symmetric tensor, and if simple tensors in decompositions are required to be symmetric, then the corresponding smallest value of r is called the symmetric rank of T . A basic result of linear algebra says that these two notions of rank agree for symmetric matrices, and an analogous statement for tensors has become known as Comon's conjecture.Conjecture 1. The symmetric rank of a symmetric tensor equals its rank.Symmetric tensors arise naturally in different applications, so it is an important problem to compute the symmetric ranks and corresponding decompositions, see [2,6,7,11] and references therein. Conjecture 1 received a considerable amount of attention in recent publications, which include [1,3,4,5,6,8,9,11,13,15], but it has been proved in several special cases only. The general form of Conjecture 1 remained open, and the aim of this paper is to construct a counterexample.

The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is the work [42] by Strassen, who presented an O(n ln 7/ ln 2 ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [43] published in 1973, Strassen asked whether the multiplicative complexity of the union of two bilinear systems depending on different variables is equal to the sum of the multiplicative complexities of both systems. A stronger version of this problem was proposed in the 1981 paper [19] by Feig and Winograd, who asked whether any optimal algorithm that computes such a pair of bilinear systems must compute each system separately. These questions became known as the direct sum conjecture and strong direct sum conjecture, respectively, and they were attracting a notable amount of attention during the four decades. As Feig and Winograd wrote, "either a proof of, or a counterexample to, the direct sum conjecture will be a major step forward in our understanding of complexity of systems of bilinear forms."

Let G be a semigroup. The vertices of the power graph P(G) are the elements of G, and two elements are adjacent if and only if one of them is a power of the other. We show that the chromatic number of P(G) is at most countable, answering a recent question of Aalipour et al.