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The distribution of rational points on varieties defined over a finite field

Published online by Cambridge University Press:  26 February 2010

Gerald Myerson
Gerald Myerson
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, New York, U.S.A.
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Extract

Let h1(x1, …, xn), …, hs(x1, …, xn) be polynomials with integer coefficients. We give conditions on these polynomials which guarantee the existence, for all sufficiently large primes p, of small solutions to the system of congruences

Previous investigations of this problem include those of Mordell [10], Chalk and Williams [5], and Smith [14]. Smith's main result, which encompasses the other results, can be stated as follows.

Type
Research Article
Information
Mathematika , Volume 28 , Issue 2 , December 1981 , pp. 153 - 159
Copyright
Copyright © University College London 1981

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References

1.Bombieri, E.. “On exponential sums in finite fields”, Amer. J. Math., 88 (1966), 71–105.CrossRefGoogle Scholar
2.Bombieri, E.. “On exponential sums in finite fields, II”, Inventions Math., 47 (1978), 29–39.CrossRefGoogle Scholar
3.Chalk, J. H. H.. “The number of solutions of congruences in incomplete residue systems”, Canad. J. Math., 15 (1963), 291–296.Google Scholar
4.Chalk, J. H. H.. “The Vinogradov-Mordell-Tietäväinen Inequalities”, Indag. Math., 42 (1980), 367–374.Google Scholar
5.Chalk, J. H. H. and Williams, K. S.. “The distribution of solutions of congruences”, Mathematika, 12 (1965), 176–192.CrossRefGoogle Scholar
6.Chalk, J. H. H. and Williams, K. S.. “The distribution of solutions of congruences, corrigendum and addendum”, Mathematika, 16 (1969), 98–100.CrossRefGoogle Scholar
7.Deligne, P.. “La conjecture de Weil I”, Publ. Math. IHES, 43 (1974), 273–307.Google Scholar
8.Deligne, P. et al. Séminaire de géometrie algébrique du Bois-Marie SGA . In Cohomologie Étale, Lecture Notes in Math., 569 (Springer, Berlin, 1977).Google Scholar
9.Lang, S. and Weil, A.. “Number of points on varieties in finite fields”, Amer. J. Math., 76 (1954), 819–827.Google Scholar
10.Mordell, L. J.. “On the number of solutions in incomplete residue sets of quadratic congruences”, Arch. der Math., 8 (1957), 153–157.CrossRefGoogle Scholar
11.Myerson, G.. “A combinatorial problem in finite fields, II”, Quart. J. Math., 31 (1980), 219–231.Google Scholar
12.Schmidt, W.. Equations over Finite Fields, Lecture Notes in Math., 536 (Springer, Berlin, 1976).Google Scholar
13.Serre, J.-P.. “Majorations de sommes exponentielles”, Soc. Math. France, Astérisque, 41-42 (1977), 111–126.Google Scholar
14.Smith, R.. “The distribution of rational points on hypersurfaces denned over a finite field”, Mathematika, 17 (1970), 328–332.CrossRefGoogle Scholar
15.Spackman, K. W.. “On the number and distribution of simultaneous solutions to diagonal congruences”, Canad. J. Math., 33 (1981), 421–436.CrossRefGoogle Scholar
16.Tietäväinen, A.. “On the solvability of equations in incomplete finite fields”, Annales Universitatis Turkuensis, Series A, 1, 102 (1967), 1–13.Google Scholar