The key insights and highlights of the content are:
The authors introduce the concept of a "composition basis" for the linear span of the Kronecker power map Kd,q, which provides an explicit basis for expressing the Kronecker powers of tensors in Fd ⊗Fd ⊗Fd.
Using the composition basis, the authors construct an explicit sequence of tensors Ud = (Ud,q : q = 1, 2, ...) such that σ(Ud) = σ(d), where σ(d) is the supremum of the tensor exponents over all tensors in Fd ⊗Fd ⊗Fd.
The authors also construct an explicit sequence of tensors Td such that σ(Td) = σ(Td), where Td is the space of tight tensors in Fd ⊗Fd ⊗Fd. This provides a universal sequence for Strassen's asymptotic rank conjecture.
By diagonalizing the sequences Ud, the authors obtain an explicit sequence D = (Dd : d = 1, 2, ...) that universally captures the limit of the worst-case tensor exponent limd→∞σ(d), addressing the extended asymptotic rank conjecture.
The authors show that the absence of low-degree polynomial equations vanishing on tensors of low asymptotic rank implies upper bounds on the asymptotic rank, providing a new technique for bounding asymptotic rank.
Overall, the paper presents explicit universal constructions of tensors that characterize the worst-case asymptotic rank behavior, with implications for resolving long-standing conjectures in computational complexity theory.
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