Structure vs. randomness for bilinear maps

We prove that the slice rank of a 3-tensor (a combinatorial notion introduced by Tao in the context of the cap-set problem), the analytic rank (a Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank (a recently introduced algebro-geometric notion) are all equivalent up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bililnear map, and on the separation between computing a bilinear map exactly and on average. Our result settles open questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal., 2019] for 3-tensors.

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