On almost primes solutions to forms of odd degrees in many variables
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909025436696576 |
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| author | Magyar, Akos |
| author_facet | Magyar, Akos |
| contents | Let $\mathcal{F}=\{f_1,\ldots,f_R\}$ be a family of forms of odd degrees at most $d$ in $s$ variables. We study the solutions to the system $f_1(\mathbf{x})=\ldots=f_R(\mathbf{x})=0$ of the form $x_i=y_ip_i$ with $|y_i|\leq Y_\mathcal{F}$ and $p_i$ being a prime for all $i\in [s]$ inside the box $[-N,N]^s$, for large $N$. We show that if the number of variables $s$ is sufficiently large with respect to the parameters $R$ and $d$, then there are at least $C_\mathcal{F} N^{s-D}/(\log\,N)^s$ such solutions for some constants $C_\mathcal{F}>0$ and $D\in\mathbb{N}$, with $D$ depending only on the initial parameters $R$ and $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06942 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On almost primes solutions to forms of odd degrees in many variables Magyar, Akos Number Theory 11D72, 11P55 Let $\mathcal{F}=\{f_1,\ldots,f_R\}$ be a family of forms of odd degrees at most $d$ in $s$ variables. We study the solutions to the system $f_1(\mathbf{x})=\ldots=f_R(\mathbf{x})=0$ of the form $x_i=y_ip_i$ with $|y_i|\leq Y_\mathcal{F}$ and $p_i$ being a prime for all $i\in [s]$ inside the box $[-N,N]^s$, for large $N$. We show that if the number of variables $s$ is sufficiently large with respect to the parameters $R$ and $d$, then there are at least $C_\mathcal{F} N^{s-D}/(\log\,N)^s$ such solutions for some constants $C_\mathcal{F}>0$ and $D\in\mathbb{N}$, with $D$ depending only on the initial parameters $R$ and $d$. |
| title | On almost primes solutions to forms of odd degrees in many variables |
| topic | Number Theory 11D72, 11P55 |
| url | https://arxiv.org/abs/2605.06942 |