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Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Multiplicative Complexity of Commutative Group Algebras over Algebraically Closed Fields
, 2007
"... We determine structure and multiplicative complexity of commutative group algebras over algebraically closed fields. Commutative group algebra A of dimension n over algebraically closed field is isomorphic to B t, where B is a superbasic algebra of minimal rank (see [5] for definition), and t is max ..."
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We determine structure and multiplicative complexity of commutative group algebras over algebraically closed fields. Commutative group algebra A of dimension n over algebraically closed field is isomorphic to B t, where B is a superbasic algebra of minimal rank (see [5] for definition), and t is maximal that t  n, p ∤ t. Multiplicative and bilinear complexity of A equal to 2n − t. 1
Algebras of minimal multiplicative complexity
"... Abstract—We prove that an associative algebra A has minimal rank if and only if the Alder–Strassen bound is also tight for the multiplicative complexity of A, that is, the multiplicative complexity of A is 2 dimA − tA where tA denotes the number of maximal twosided ideals of A. This generalizes a r ..."
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Abstract—We prove that an associative algebra A has minimal rank if and only if the Alder–Strassen bound is also tight for the multiplicative complexity of A, that is, the multiplicative complexity of A is 2 dimA − tA where tA denotes the number of maximal twosided ideals of A. This generalizes a result by E. Feig who proved this for division algebras. Furthermore, we show that if A is local or superbasic, then every optimal quadratic computation for A is almost bilinear. Keywordsalgebraic complexity theory; complexity of bilinear problems; associative algebra. I.