On the number of solutions of a Diophantine equation with symmetric entries
type: Item
instance of: scholarly article
title: On the number of solutions of a Diophantine equation with symmetric entries
zbMATH Open document ID: 1135.11012
published in: Journal of Number Theory
publication date: 2007-06-06T00:00:00Z
review text: 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 \(f(x_1+1)-f(x_1)>f(x_0+1)-f(x_0)>0\).
reviewed by: Nikos Tzanakis
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 5161942
zbMATH Keywords: convex sequences related to Diophantine equations
author: Moubariz Z. Garaev
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.09.018
OpenAlex ID: W2086551468
cites work: On lower bounds for the \(L_1\)-norm of exponential sums
cites work: Upper bounds for the number of solutions of a Diophantine equation
cites work: Combinatorial complexity of convex sequences
cites work: An estimate of the \(L_1\)-norm of an exponential sum
DOI: 10.1016/J.JNT.2006.09.018
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description: scientific article; zbMATH DE number 5161942