On a certain nonary cubic form and related equations
type: Item
instance of: scholarly article
title: On a certain nonary cubic form and related equations
zbMATH Open document ID: 0847.11052
DOI: 10.1215/S0012-7094-95-08023-5
published in: Duke Mathematical Journal
publication date: 1996-06-24T00:00:00Z
review text: Let \(S_3(P; 1, 3)\) denote the number of solutions of the system \[ x^3+ y^3+ z^3= X^3+ Y^3+ Z^3,\quad x+ y+ z= X+ Y+ Z, \] \[ 1\leq x, y, z, X, Y, Z\leq P. \] \textit{L.-K. Hua} [Additive theory of prime numbers, Am. Math. Soc. (1965; Zbl 0192.39304)] showed \(S_3(P; 1,3)\ll P^3\log^9 P\), this being of importance in connection with Weyl's inequality an d Hua's lemma. By elementary but intricate methods the authors establish a much more precise result \(S_3(P; 1, 3)= 6P^3+ U(P)\), where \(P^2\log^5 P\ll U(P)\ll P^2\log^5 P\). Here, the term \(6P^3\) arises from the ``trivial'' solutions in which \(x\), \(y\), \(z\) are a permutation of \(X\), \(Y\), \(Z\). They suggest that their methods might lead to an asymptotic formula for the number \(U(P)\) of non-trivial solutions, but this has not yet been done, because of the further complications that would appear likely to be involved. Further, let \(S_s(P; k)\) denote the number of solutions of the system \(\sum^s_{i= 1} (x^{k_j}_i- y^{k_j}_i)= 0\) \((j= 1,\dots, t)\) with \(1\leq x_i\), \(y_i\leq P\). They record relatively weaker upper bounds (of the form \(\ll P^{s- \delta}\) with explicit \(\delta= \delta_s> 0\)) for the quantities analogous to \(U(P)\) in the two cases \(k= \{1, 2,\dots, s- 2, s\}\), \(k= \{1, 2,\dots, s- 1\}\), the last of which is a case of Vinogradov's mean value theorem.
reviewed by: George Greaves
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 871841
zbMATH Keywords: nonary cubic form
zbMATH Keywords: number of solutions
zbMATH Keywords: Weyl's inequality
zbMATH Keywords: Hua's lemma
zbMATH Keywords: Vinogradov's mean value theorem
Wikidata QID: Q106135758
author: Robert C. Vaughan
author: Trevor D. Wooley
MaRDI profile type: Publication
cites work: On the number of rational points of bounded height on algebraic varieties
cites work: A remarkable identity found in Ramanujan's third notebook
cites work: Forms in many variables
cites work: The asymptotic formula in Waring's problem
cites work: A geometric approach to equal sums of fifth powers
cites work: A Geometric Approach to Equal Sums of Sixth Powers
cites work: A complete determination of the zeros of weight-1 6j coefficients
cites work: On the Hasse principle for cubic surfaces
cites work: Symmetric Diophantine systems
cites work: Density of integer points on affine homogeneous varieties
cites work: Rational points of bounded height on Fano varieties
cites work: Q5838388
cites work: On the representation of a number as a sum of two fourth powers
cites work: Some Diophantine equations with almost all solutions trivial
cites work: Q5772619
cites work: Weyl's Inequality, Hua's Inequality, and Waring's Problem
cites work: On the representations of a number as the sum of two cubes
cites work: On the representation of a number as the sum of two h-th powers
cites work: On the numbers that are representable as the sum of two cubes.
cites work: On Another Sieve Method and the Numbers that are a Sum of Two h th Powers
cites work: On Waring's problem
cites work: On some topics connected with Waring's problem.
cites work: Q5587215
cites work: Singular matrices and a uniform bound for congruence groups of \(SL_ n(\mathbb{Z})\)
cites work: Number theory III: Diophantine geometry
cites work: Q4694970
cites work: Q3792745
cites work: The density of integer points on homogeneous varieties
cites work: Q4170817
cites work: Q3735790
cites work: Q3903082
full work available at URL: s0012-7094-95-08023-5
OpenAlex ID: W2013899154
Recommended article: Some Diophantine equations with almost all solutions trivial
Recommended article: A New Iterative Method in Waring's Problem II
Recommended article: The paucity problem for simultaneous quadratic and biquadratic equations
Recommended article: Q4351616
Recommended article: Q3792745
Recommended article: Rational lines on cubic hypersurfaces
Recommended article: Some remarks on Vinogradov's mean value theorem and Tarry's problem
Recommended article: VINOGRADOV SYSTEMS WITH A SLICE OFF
Recommended article: Sums of cubes of square-free numbers. II
Recommended article: The asymptotic formula for localized solutions in Waring's problem and approximations to Weyl sums
description: scientific article; zbMATH DE number 871841