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  1. Abstract

    Let$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$denote the matrix multiplication tensor (and write$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$), and let$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$denote the determinant polynomial considered as a tensor. For a tensorT, let$\underline {\mathbf {R}}(T)$denote its border rank. We (i) give the first hand-checkable algebraic proof that$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$, (ii) prove$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$and$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$M_{\langle 2\rangle }$, (iii) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$, (iv) prove$\underline {\mathbf {R}}(\operatorname {det}_3)=17$, improving the previous lower bound of$12$, (v) prove$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$for all$\mathbf {n}\geq 25$, where previously only$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$was known, as well as lower bounds for$4\leq \mathbf {n}\leq 25$, and (vi) prove$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$for all$\mathbf {n} \ge 18$, where previously only$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

    The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

     
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  2. Free, publicly-accessible full text available December 1, 2024
  3. We determine defining equations for the set of concise tensors of minimal border rank in Cm⊗Cm⊗Cm when m = 5 and the set of concise minimal border rank 1∗-generic tensors when m = 5, 6. We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case m = 5. Our proofs utilize two recent developments: the 111-equations defined by Buczy´nska–Buczy´nski and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in C5⊗C5⊗C5. 
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  4. This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical computer science, and on the other hand touches on many beautiful topics in algebraic geometry such as classical and recent results on equations for secant varieties (e.g., via vector bundle and representation-theoretic methods) and the geometry and deformation theory of zero dimensional schemes. 
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