论文标题
高级多项式的等式分布，变量仅限于$ \ mathbb {f} _p $的子集
Equidistribution of high-rank polynomials with variables restricted to subsets of $\mathbb{F}_p$
论文作者
论文摘要
令$ p $为prime，让$ s $为$ \ mathbb {f} _p $的非空子集。 Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \ Mathbb {f} _p^n \ rightArrow \ Mathbb {f} _p $，在$ s^n $上消失，使得多项式$ p-p_0 $具有有限的等级。我们的论点使用了两个黑匣子：具有较高分区排名的张量具有很高的分析等级，并且具有高基本分区等级的张量具有很高的分区等级。
Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank and that a tensor with high essential partition rank has high disjoint partition rank.
