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arXiv · 2507.14414

On weighted multilinear polynomial averages in finite fields

Guo-Dong Hong
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Abstract

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.

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Guo-Dong Hong. 2025-07-18. On weighted multilinear polynomial averages in finite fields. https://arxiv.org/abs/2507.14414

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