On almost primes solutions to forms of odd degrees in many variables

TL;DR

This paper investigates solutions to systems of odd-degree forms in many variables, showing that under certain conditions, many solutions of a specific almost prime form exist within a large box.

Contribution

It establishes the existence of numerous almost prime solutions to systems of odd-degree forms in many variables, extending previous results in the field.

Findings

Existence of at least C_F N^{s-D}/(log N)^s solutions under certain conditions.

Solutions are of the form x_i = y_i p_i with p_i prime and |y_i| bounded.

Number of solutions grows polynomially with N, modulated by a logarithmic factor.

Abstract

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 $d$.