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arXiv · 1905.07355

A gap in the slice rank of $k$-tensors

Simone Costa·Marco Dalai
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Abstract

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method. In this paper, building upon a method presented by Tao and Sawin, it is proved that the asymptotic slice rank of any $k$-tensor in any field is either $1$ or at least $k/(k-1)^{(k-1)/k}$. This provides evidence that straight-forward application of the method cannot give useful results in certain problems for which non-trivial exponential bounds are already known. An example, actually a motivation for starting this work, is the problem of bounding the size of trifferent sets of sequences, which constitutes a long-standing open problem in information theory and in theoretical computer science.

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Simone Costa, Marco Dalai. 2019-05-17. A gap in the slice rank of $k$-tensors. https://arxiv.org/abs/1905.07355

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