Paucity phenomena for polynomial products
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | |
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| _version_ | 1866929461314715648 |
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| author | Wang, Victor Y. Xu, Max Wenqiang |
| author_facet | Wang, Victor Y. Xu, Max Wenqiang |
| contents | Let $P(x)\in \mathbb{Z}[x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_1, \dots, x_k, y_1, \dots, y_k)\in [N]^{2k}$ to the equation \[
\prod_{1\le i \le k} P(x_i) = \prod_{1\le j \le k} P(y_j)\neq 0 \] (for any $k\ge 1$) is asymptotically $k!N^{k}$ as $N\to +\infty$. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums $\frac{1}{\sqrt{N}}\sum_{n\le N}f(P(n))$ match standard complex Gaussian moments as $N\to +\infty$, where $f$ is the Steinhaus random multiplicative function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02908 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Paucity phenomena for polynomial products Wang, Victor Y. Xu, Max Wenqiang Number Theory Probability Let $P(x)\in \mathbb{Z}[x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_1, \dots, x_k, y_1, \dots, y_k)\in [N]^{2k}$ to the equation \[ \prod_{1\le i \le k} P(x_i) = \prod_{1\le j \le k} P(y_j)\neq 0 \] (for any $k\ge 1$) is asymptotically $k!N^{k}$ as $N\to +\infty$. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums $\frac{1}{\sqrt{N}}\sum_{n\le N}f(P(n))$ match standard complex Gaussian moments as $N\to +\infty$, where $f$ is the Steinhaus random multiplicative function. |
| title | Paucity phenomena for polynomial products |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2211.02908 |