Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T21:26:07.970Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of solutions of congruences

Published online by Cambridge University Press:  26 February 2010

J. H. H. Chalk  and
K. S. Williams
J. H. H. Chalk
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada.
K. S. Williams
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada.
Get access

Extract

Let p be an odd prime and denote by [p], the finite field of residue classes, mod p. In Euclidean n-space, let n denote the lattice of points x = (x1, …, xn) with integral coordinates and C = C(n, p), the set of points of n satisfying

Type
Research Article
Information
Mathematika , Volume 12 , Issue 2 , December 1965 , pp. 176 - 192
Copyright
Copyright © University College London 1965

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.Birch, B. J. and Lewis, D. J., “p-adic forms”, J. Indian Math. Soc., 23 (1959), 11–32.Google Scholar
2.Birch, B. J., “Forms in many variables”, Proc. Roy. Soc. (A), 265 (1962), 245–263. (See [5] p. 652.)Google Scholar
3.Chalk, J. H. H., “The number of solutions of congruences in incomplete residue systems”, Canadian J. of Math., 15 (1963), 291–296.CrossRefGoogle Scholar
4.Chevelley, C., “Démonstration d'une hypothèse de M. Artin”, Abh. Math. Sem. Hamburg., 11 (1935), 73–75.CrossRefGoogle Scholar
5.Davenport, H. and Lewis, D. J., “Exponential sums in many variables”, American J. of Math., 84 (1962), 649–665.CrossRefGoogle Scholar
6.Lang, S. and Weil, A., “Number of points of varieties in finite fields”, American J. of Math., 76 (1954), 819–827.CrossRefGoogle Scholar
7.Min, S. H., “On systems of algebraic equations and certain multiple exponential sums”. Quart. J. of Math., (Oxford), 18 (1947), 133–142.Google Scholar
8.Mordell, L. J., “On the number of solutions in incomplete residue sets of quadratic congruences”, Archiv der Math., 8 (1957), 153–157.CrossRefGoogle Scholar
9.Mordell, L. J., “Incomplete exponential sums and incomplete residue systems for congruences”, MexocjTOBaqKHK MaTeMaTHiecKHii )KypHan (Czech. Math. J.), 14 (1964), 235–242.CrossRefGoogle Scholar
10.Perron, O., Algebra I (2nd. ed. Berlin 1932).Google Scholar
11.Vinogradov, I. M., Elements of number theory, (Dover 1954), Chap. V, problem 12a, p. 102.Google Scholar
12.Weil, A., Foundations of algebraic geometry (New York), Amer. Math. Soc. Colloq. Pub., 29 (1946).CrossRefGoogle Scholar