On weighted multilinear polynomial averages in finite fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909696349175808 |
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| author | Hong, Guo-Dong |
| author_facet | Hong, Guo-Dong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On weighted multilinear polynomial averages in finite fields Hong, Guo-Dong Number Theory 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. |
| title | On weighted multilinear polynomial averages in finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.14414 |