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On weighted multilinear polynomial averages in finite fields

On weighted multilinear polynomial averages in finite fields

来源:Arxiv_logoArxiv
英文摘要

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 Teräväinen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of $\mathbb{F}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(φ(y))v_1,\cdots, \textbf{x}+ P_k(φ(y))v_k, \] 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 $φ(y) \in \mathbb{Q}(y)$ is a nonzero rational function.

Guo-Dong Hong

数学

Guo-Dong Hong.On weighted multilinear polynomial averages in finite fields[EB/OL].(2025-07-18)[2025-08-18].https://arxiv.org/abs/2507.14414.点此复制

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