Simultaneous diagonal congruences
type: Item
instance of: scholarly article
title: Simultaneous diagonal congruences
zbMATH Open document ID: 0643.10011
DOI: 10.1016/0022-314X(88)90092-3
author: Lewis Low
author: Jane Pitman
author: Alison Wolff
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Considering R diagonal integral forms of degree k in n variables, \[ F_ i({\mathfrak x})=F_ i=a_{i1}x^ k_ 1+...+a_{n1}x^ k_ n\quad (i=1,...,R), \] \textit{H. Davenport} and \textit{D. J. Lewis} [Philos. Trans. R. Soc. Lond., Ser. A 264, 557-595 (1969; Zbl 0207.353)] showed that the conditions \(n\geq [9R^ 2k \log 3Rk]\) for odd k, and \(n\geq [48R^ 2k^ 3 \log 3Rk^ 2]\) for even k, ensure the existence of a primitive solution of the congruences \[ (1)\quad F_ 1\equiv...\equiv F_ R\equiv 0 (mod p^ s) \] for all prime powers \(p^ s\) if the forms are suitably normalized. Furthermore, they showed that these conditions on n also guarantee the existence of a non-trivial p-adic solution of the equations \((2)\quad F_ 1=...=F_ R=0.\) In the present paper the authors first present conditions for partitioning the coefficient matrix \([a_{ij}]\) and then obtain explicitly a lower bound for the number of solutions of (1) with odd k. As a consequence they prove that if \(n\geq 2R^ 2k \log k\) for large odd k or \(n\geq [48Rk^ 3 \log 3Rk^ 2]\) for any \(k\geq 3\), then there exists a non-trivial p-adic solution of (2) for each prime p. An interesting discussion on the relationship of the authors' methods to the work of Davenport and Lewis [ibid.] and \textit{W. M. Schmidt} [J. Number Theory 19, 63-80 (1984; Zbl 0541.10024)] is given in the paper.
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 11E76
zbMATH DE Number: 4047801
zbMATH Keywords: simultaneous congruences
zbMATH Keywords: simultaneous equations
zbMATH Keywords: partitionable matrices
zbMATH Keywords: diagonal integral forms
zbMATH Keywords: lower bound
zbMATH Keywords: number of solutions
zbMATH Keywords: p-adic solution
reviewed by: Ming-Chit Liu
MaRDI profile type: Publication
cites work: Q3050437
cites work: Q5521595
cites work: Q5730231
cites work: Pairs of additive equations
cites work: Simultaneous equations of additive type
cites work: Minimum partition of a matroid into independent subsets
cites work: Q3800137
cites work: Bounds for solutions of diagonal inequalities
cites work: Simultaneous p-adic zeros of quadratic forms
cites work: The solubility of certain p-adic equations
cites work: On a problem of Chowla and Shimura
cites work: Q3832660
full work available at URL: 0022-314x(88)90092-3
OpenAlex ID: W2087434374
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description: scientific article; zbMATH DE number 4047801