On the number of solutions of a Diophantine equation with symmetric entries
Fix a certain strictly increasing real valued function \(f\) and denote by \(J_d(N)\) the number of integer solutions \((x_1,\dots,x_d,y_1,\dots,y_d)\) of the equation \[ \sum_{i=1}^s f(x_i)= \sum_{i=1}^s f(y_i) \] such that all \(x_i\)'s and all \(y_i\)'s belong to the interval \([1,N]\). There is a result of Konyagin (2000) for a general strictly convex sequence \(f(n)\), saying that \(J_2(N)\ll N^{5/2}\), where the symbol \(\ll\), here and below, depends on \(d\) and other fixed constants. For \(d>2\), \textit{A. Iosevich, S. Konyagin, M. Rudnev} and \textit{V. Ten} proved in [Discrete Comput. Geom. 35, No. 1, 143--158 (2006; Zbl 1116.11011)] that \(J_d(N)\ll N^{2d-2+2^{1-d}}\) using higher dimensional incidence theory in combination with an inductive procedure. In the present paper the author introduces a different approach, independent from incidence theory and proves that \(J_d(N)\leq 100^dN^{2d-2+d^{-1}}\), provided that \(f\) satisfies the following condition: any positive integers \(x_0,x_1\) satisfying \(x_1-x_0\geq [N^{1/(2d)}]\), also satisfy \(f(x_1+1)-f(x_1)>f(x_0+1)-f(x_0)>0\).
convex sequences related to Diophantine equations
scientific article; zbMATH DE number 5161942