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Algebra |
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Authors: Yaroslav Shitov
The substitution method of tensor rank computation is a higher dimensional analogue of Gaussian elimination, and it builds on the fact that the removal of a rank one slice s and a subsequent addition of arbitrary scalar multiples of s to all other slices of the same direction decreases the minimum rank exactly by one. We explain how to embed an initial tensor T to a larger linear space and replace its higher rank slice g by a family f of rank one slices in the new space so that the substitutions performed with respect to g in every direction of T have the same effect on the minimum rank as the corresponding substitutions with respect to f. We present several applications, which include a resolution of the well known and widely studied direct sum conjecture for Waring ranks and a strong form of counterexamples to Strassen’s conjecture.
Comments: 85 Pages.
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[v1] 2023-07-19 03:16:53
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