Almost prime solutions to diophantine systems of high rank

TL;DR

This paper investigates the distribution of solutions to high-rank Diophantine systems where certain linear forms take on values with a limited number of prime factors, extending Birch's results to almost prime solutions.

Contribution

It establishes the existence of expected numbers of almost prime solutions to high-rank Diophantine systems under conditions similar to those for integer solutions.

Findings

Proves the existence of almost prime solutions under high-rank conditions.

Extends Birch's results from integer solutions to almost prime solutions.

Provides quantitative estimates for the number of such solutions.

Abstract

Let $\F$ be a family of $r$ integral forms of degree $k\geq 2$ and $\LL=(l_1,\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\x=(x_1,...,x_n)$. We study the number of solutions $\x\in[1,N]^n$ to the diophantine system $\F(\x)=\vv$ under the restriction that $l_i(\x)$ has a bounded number of prime factors for each $1\leq i\leq m$. We show that the system $\F$ have the expected number of such "almost prime" solutions under similar conditions as was established for existence of integer solutions by Birch.