Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-29T07:18:00.871Z Has data issue: false hasContentIssue false

Simultaneous diagonal p–adic equations

Published online by Cambridge University Press:  26 February 2010

I. D. Meir
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH. Tricorder Technology plc, The Long Room, Coppermill Lock, Summerhouse Lane, Harefield, Middlesex UB9 6JA.
Get access

Abstract

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k2r + 2. In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r2k2+(2/(c − 2)) if r ≠ 1 and p > k2+(2/(c − 1)) if r = 1.

MSC classification

Type
Research Article
Copyright
Copyright © University College London 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Atkinson, O. D., Brüdern, J. and Cook, R. J.. Simultaneous additive congruences to a large prime modulus. Mathematika, 39 (1992), 19.Google Scholar
2.Aigner, M.. Combinatorial Theory (Springer, 1979).Google Scholar
3.Davenport, H.. Analytic Methods for Diophantine Equations and Diophantine Inequalities (Campus Publishers, Ann Arbor, 1963).Google Scholar
4.Davenport, H. and Lewis, D. J.. Simultaneous equations of additive type. Phil. Trans. Roy. Soc. London, 264A (1969), 557595.Google Scholar
5.Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge University Press, 1952).Google Scholar
6.Schmidt, W. M.. Equations over Finite Fields, An Elementary Approach, Lecture Notes in Mathematics, 536 (Springer, 1976).Google Scholar