2021/11/11 by Shuntaro Yamagishi, Yamagishi, Shuntaro
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2111.06122
openalex publication_date 2021/11/11 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Let F ∈ ℤ[x1, …, xn] be a homogeneous form of degree d ≥ 2, and let VF^* denote the singular locus of the affine variety V(F) = \ z ∈ \mathbbAnℂ: F(z) = 0 \. In this paper, we prove the existence of integer solutions with prime coordinates to the equation F(x1, …, xn) = 0 provided F satisfies suitable local conditions and n - dim VF^* ≥ 7 d (2d-1) 4d + 4 (d-1) (12d - 1) 2d + 12d. The result is obtained by using the identity Λ= μ* log for the von Mangoldt function and optimizing various parts of the argument in the author's previous work, which made use of the Vaughan identity and required n - dim VF^* ≥ 28 34 52 d3 (2d-1)2 4d.