Some remarks on Vinogradov's mean value theorem and Tarry's problem
type: Item
instance of: scholarly article
title: Some remarks on Vinogradov's mean value theorem and Tarry's problem
zbMATH Open document ID: 0881.11043
DOI: 10.1007/BF01320189
author: Trevor D. Wooley
published in: Monatshefte für Mathematik
publication date: 1998-02-16T00:00:00Z
full work available at URL: https://eudml.org/doc/178747
review text: Let \(k\) be a positive integer and \(W(k)\) denote the least number \(s\) for which the system of equations \[ \sum^s_{i=1} x_i^j= \sum_{i=1}^s y_i^j, \qquad j=1,\dots, k, \] has a solution with \(\sum_{i=1}^s x_i^{k+1}\neq \sum_{i=1}^s y_i^{k+1}\). With the help of his recent results in [\textit{T. D. Wooley}, Mathematika 39, 379--399 (1992; Zbl 0769.11036); J. Am. Math. Soc. 7, 221--245 (1994; Zbl 0786.11053) and in Number theory with an emphasis on the Markoff spectrum, Lect. Notes Pure Appl. Math. 147, 317--321 (1993; Zbl 0791.11014)], the author proves that for large \(k\) \[ W(k)\leq \tfrac12 k^2(\log k+\log\log k+O(1)), \] and that for each \(\varepsilon>0\) there is a real \(K(\varepsilon)\) with the property that for each \(K\geq K(\varepsilon)\) there exists a \(k\) in the interval \([K, K^{4/3+\varepsilon}]\) with \[ W(k)\leq \tfrac12 k(k+1)+1. \] He also shows that the least \(s\) for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfies \(s\leq k^2(\log k+O(\log\log k))\).
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 1051737
zbMATH Keywords: Vinogradov's mean value theorem
zbMATH Keywords: Tarry's problem
zbMATH Keywords: system of symmetric diagonal equations
zbMATH Keywords: asymptotic formula
reviewed by: Ming-Chit Liu
MaRDI profile type: Publication
cites work: Q3856819
cites work: Improvement of a Result of Wright
cites work: Q5587215
cites work: Q3903082
cites work: A new iterative method in Waring's problem
cites work: Large improvements in Waring's problem
cites work: On Vinogradov's mean value theorem
cites work: Q3140909
cites work: Quasi-Diagonal Behaviour in Certain Mean Value Theorems of Additive Number Theory
cites work: ON TARRY'S PROBLEM (I)
cites work: Equal sums of like powers
cites work: The Prouhet-Lehmer Problem
OpenAlex ID: W1986068415
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description: scientific article; zbMATH DE number 1051737