双线性映射的结构与随机性
Structure vs. Randomness for Bilinear Maps
摘要：
我们证明了3-张量的切片秩（由陶在帽集问题的背景下引入的组合概念），分析秩（由Gowers和Wolf引入的傅立叶理论概念）和几何秩（由Kopparty，Moshkovitz和Zuiddam引入的代数几何概念）都等于一个绝对常数。作为一个推论，我们得到强权衡的算术复杂性的有偏双线性映射，并计算一个双线性映射之间的分离，准确和平均。我们的结果解决了Haramaty和Shpilka [STOC 2010]以及Lovett [Discrete Anal. [2019] 3-tensors。
英文摘要：
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 (an algebro-geometric notion introduced by Kopparty, Moshkovitz, and Zuiddam) are all equal up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bilinear 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
