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 Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] 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 [10] 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.’ The modern formulation of this conjecture is based on a natural representation of a bilinear system as a three-dimensional tensor, that is, an array of elements T (i|j|k) taken from a field F , where the triples (i, j, k) run over the Cartesian product of finite indexing sets I, J,K. A tensor T is called decomposable if T = a⊗b⊗c (which should be read as T (i|j|k) = aibjck), for some vectors a ∈ FI , b ∈ FJ , c ∈ FK . The rank of a tensor T , or the multiplicative complexity of the corresponding bilinear system, is the smallest r for which T can be written as a sum of r decomposable tensors with entries in F . We denote this quantity by rankF T , and we note that the rank of a tensor may change if one allows to take the entries of decomposable tensors as above from an extension of F , see [3]. Taking the union of two bilinear systems depending on disjoint sets of variables corresponds to the direct sum operation on tensors. More precisely, if T and T ′ are tensors with disjoint indexing sets I, I , J, J ,K,K , then we can define the direct sum T⊕T ′ as a tensor with indexing sets I ∪ I , J ∪ J , K ∪ K ′ such that the (I|J |K) block equals T and (I ′|J ′|K ) block equals T , and all entries outside of these blocks are zero. In other words, direct sums of tensors are a multidimensional analogue of block-diagonal matrices; a basic result of linear algebra says that the ranks of such matrices are equal to the sums of the ranks of their diagonal blocks. Strassen’s direct sum conjecture is a three-dimensional analogue of this statement.

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施特拉森直和猜想的反例

双线性形式系统的乘法复杂性（尤其是著名的快速矩阵乘法问题）是现代计算理论的一个重要研究领域。关于该主题的基础论文之一是 Strassen 的工作 [20]，其中包含用于两个 n×n 矩阵相乘的 O(n 7/ ln ) 算法。在随后发表于 1973 年的论文 [21] 中，Strassen 询问取决于不同变量的两个双线性系统的并集的乘法复杂度是否等于两个系统的乘法复杂度之和。Feig 和 Winograd 在 1981 年的论文 [10] 中提出了这个问题的更强版本，他们询问是否有任何计算这样一对双线性系统的最佳算法必须单独计算每个系统。这些问题分别被称为直和猜想和强直和猜想，并在这四个十年中引起了极大的关注。正如 Feig 和 Winograd 所写，“直接和猜想的证明或反例将是我们理解双线性形式系统复杂性的重要一步。” 该猜想的现代表述基于双线性系统作为三维张量的自然表示，即取自场 F 的元素阵列 T (i|j|k)，其中三元组 (i, j, k) 在有限索引集 I, J,K 的笛卡尔积上运行。如果 T = a⊗b⊗c（应读作 T (i|j|k) = aibjck），则称张量 T 为可分解的，对于某些向量 a ∈ FI , b ∈ FJ , c ∈ FK 。张量 T 的秩，或相应双线性系统的乘法复杂度，是最小的 r，其中 T 可以写为 r 个可分解张量的总和，其条目为 F 。我们用 rankF T 表示这个量，我们注意到如果允许从 F 的扩展中获取上述可分解张量的条目，张量的等级可能会改变，参见 [3]。根据不相交的变量集取两个双线性系统的并集对应于张量的直接求和运算。更准确地说，如果 T 和 T ' 是具有不相交索引集 I, I , J, J ,K,K 的张量，那么我们可以将直和 T⊕T ' 定义为具有索引集 I ∪ I , J ∪ J 的张量, K ∪ K ′ 使得 (I|J |K) 块等于 T 并且 (I ′|J ′|K ) 块等于 T ，并且这些块之外的所有条目都为零。换句话说，张量的直接和是块对角矩阵的多维模拟；线性代数的一个基本结果是，这些矩阵的秩等于它们对角块的秩之和。施特拉森的直和猜想是这个陈述的三维模拟。