Exact border subranks and tight bounds are determined for k-fold matrix multiplication and several other algebra structure tensors at all orders, together with a proof that degeneration propagates from higher to lower order.
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The authors define divisible weighted projective spaces, give sharp bounds for minimal-degree non-degenerate subvarieties therein, and develop a theory of weighted determinantal scrolls that achieve minimal degree while satisfying weighted N_p properties tied to regularity notions.
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Border subrank of higher order tensors and algebras
Exact border subranks and tight bounds are determined for k-fold matrix multiplication and several other algebra structure tensors at all orders, together with a proof that degeneration propagates from higher to lower order.
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Varieties of minimal degree in weighted projective space
The authors define divisible weighted projective spaces, give sharp bounds for minimal-degree non-degenerate subvarieties therein, and develop a theory of weighted determinantal scrolls that achieve minimal degree while satisfying weighted N_p properties tied to regularity notions.