标题： 关于有限域中的加权多线性多项式平均值 显示英文标题

标题： On weighted multilinear polynomial averages in finite fields

摘要： 我们研究有限域中的加权多线性多项式平均值。 关键要素是对应于有限域中的加权多线性多项式平均值的$u^s$范数控制，这受到 Teräväinen\cite{T24}的启发。 作为应用，我们证明了在$\mathbb{F}_p^D$的子集中的以下多维有理函数等差数列的数量的渐近公式：\[ \textbf{x}, \textbf{x}+ P_1(\varphi(y))v_1,\cdots, \textbf{x}+ P_k(\varphi(y))v_k, \]，其中$\mathbb{V}=\{v_1, \cdots, v_{k} \in \mathbb{Z}^D\}$是非零向量的集合，$\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{Z}[y]\}$是具有零常数项的线性无关多项式的集合，$\varphi(y) \in \mathbb{Q}(y)$是一个非零有理函数。 显示英文摘要

摘要： We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Ter\"av\"ainen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of $\mathbb{F}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(\varphi(y))v_1,\cdots, \textbf{x}+ P_k(\varphi(y))v_k, \] where $\mathbb{V}=\{v_1, \cdots, v_{k} \in \mathbb{Z}^D\}$ is a collection of nonzero vectors, $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $\varphi(y) \in \mathbb{Q}(y)$ is a nonzero rational function.