2021/05/26 by Shuntaro Yamagishi, Yamagishi, Shuntaro
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2105.12435
openalex publication_date 2021/05/26 · openalex created_date 2025/10/10 · 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 ∈ ℂn: 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^* ≥ 28 34 52 d3 (2d-1)2 4d. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2014), 701-737), which required n - dim VF^* to be an exponential tower in d.