Polynomials and the exponent of matrix multiplication
Copyright policy )Polynomials and the exponent of matrix multiplication
We define tensors, corresponding to cubic polynomials, which have the same exponent $ω$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $ω$.
14 pages + appendix of 3 pages with numerical decompositions
- The University of Texas System United States
- Max Planck Society Germany
- Max Planck Institute for Informatics Germany
- University of Siena Italy
- University of Notre Dame United States
Mathematics - Algebraic Geometry, polynomials, Multilinear algebra, tensor calculus, symmetric tensors, FOS: Mathematics, Mathematics (all), Computational aspects of field theory and polynomials, exponent of matrix multiplication, 68Q17, 14N05, 14Q20, 15A69, Algebraic Geometry (math.AG), tensor rank
Mathematics - Algebraic Geometry, polynomials, Multilinear algebra, tensor calculus, symmetric tensors, FOS: Mathematics, Mathematics (all), Computational aspects of field theory and polynomials, exponent of matrix multiplication, 68Q17, 14N05, 14Q20, 15A69, Algebraic Geometry (math.AG), tensor rank
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