论文标题
分区和分析等级之间的准线性关系
Quasi-linear relation between partition and analytic rank
论文作者
论文摘要
添加剂组合学，数量理论和代数几何形状中的一个重要猜想认为，张量的分区等级和分析等级在任何有限场上都等于恒定。我们证明了猜想到对数因素。 我们的证明在很大程度上与以前的工作无关，利用递归构建的多项式身份和在零多项式集合上随机步行。我们还引入了一个新的矢量值量量概念（``本地等级''），该概念是分区和分析等级之间的桥梁，并且可能具有独立的兴趣作为分析高级多项式的工具。
An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a logarithmic factor. Our proof is largely independent of previous work, utilizing recursively constructed polynomial identities and random walks on zero sets of polynomials. We also introduce a new, vector-valued notion of tensor rank (``local rank''), which serves as a bridge between partition and analytic rank, and which may be of independent interest as a tool for analyzing higher-degree polynomials.
