Simultaneous equations and inequalities

2021/07/30 by Constantinos Poulias, Poulias, Constantinos
Mathematics · #11D72 #11D75 #11L07 #11P55 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.14543

openalex publication_date 2021/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let λ_i, μ_j be non-zero real numbers not all of the same sign and let a_i, b_k be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape \begincases | λ₁ x₁^θ+ ⋯ + λ_ℓ x_ℓ^θ+ μ₁ y₁^θ+ ⋯ + μ_m y_m^θ| lt; τ
a₁ x₁^d + ⋯ a_ℓ x_ℓ^d + b₁ z₁^d + ⋯ + b_n z_n^d =0, \endcases where d≥ 2 is an integer, θ> d+1 is real and non-integral and τ is a positive real number. For such systems we obtain an asymptotic formula for the number of positive integer solutions (x, y, z) = (x₁, …, z_n) inside a bounded box. Our approach makes use of a two-dimensional version of the classical Hardy-Littlewood circle method and the Davenport--Heilbronn--Freeman method. The proof involves a combination of essentially optimal mean value estimates for the auxiliary exponential sums, together with estimates stemming from the classical Weyl and Weyl-van der Corput inequalities.

Citations

• Homogeneous additive equations

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