Almost prime solutions to diophantine systems of high rank

2016/07/19 by Magyar, Akos, Titichetrakun, Tatchai
#11D72 #11P32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.05773

Abstract

Let \F be a family of r integral forms of degree k≥ 2 and \LL=(l₁,…,l_m) be a family of pairwise linearly independent linear forms in n variables \x=(x₁,...,x_n). We study the number of solutions \x∈[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≤ i≤ 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.

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