Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T16:15:34.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Small solutions of congruences

Published online by Cambridge University Press:  26 February 2010

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey.
Get access

Extract

Throughout the paper, let m be a natural number and let F(x1,…, xn) be a form of degree k ≥ 2 with integer coefficients, n ≥ 3. We are concerned with finding solutions of the congruence

for which x is a small non-zero integer vector. For example, in the case k = 2 it was shown by Schinzel, Schlickewei and Schmidt [11] that is a solution of (1) satisfying

provided that n is odd. This is best possible for n = 3, as we shall see later. Of course we can get an exponent (1/2) + (1/(2n – 2)) trivially for even n. I do not know how to improve on this. D. R. Heath-Brown (private communication) can improve the exponent in (2) to (l/2) + ε for n ≥ 4 and prime m > C1(ε).

Type
Research Article
Information
Mathematika , Volume 30 , Issue 2 , December 1983 , pp. 164 - 188
Copyright
Copyright © University College London 1983

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.Artin, E.. Elements of algebraic geometry (New York University, 1955).Google Scholar
2.Baker, R. C. and Harman, G.. Small Fractional Parts Of quadratic forms. Proc. Edinburgh Math. Soc., 25 (1982), 269–277.CrossRefGoogle Scholar
3.Baker, R. C. and Harman, G.. Small fractional parts of polynomials. Coll. Math. Soc. Janos Bolyai 34, Topics in Classical Number Theory (Elsevier, North Holland, Amsterdam). To appear.Google Scholar
4.Cook, R. J.. On the fractional parts of an additive form. Proc. Camb. Phil. Soc., 72 (1972), 209–212.CrossRefGoogle Scholar
5.Danicic, I.. An extension of a theorem of Heilbronn. Mathematika, 5 (1958), 30–37.CrossRefGoogle Scholar
6.Deligne, P.. La conjecture de Weil. I. Publ. Math. IHES, 43 (1974), 273–307.CrossRefGoogle Scholar
7.Harman, G.. Diophantine approximation and prime numbers. Ph.D. thesis (London, 1982).Google Scholar
8.Hartshorne, R.. Algebraic geometry (Springer, New York, 1977).CrossRefGoogle Scholar
9.Heath-Brown, D. R.. Cubic forms in 10 variables. Proc. London Math. Soc., to appear.Google Scholar
10.Myerson, G.. The distribution of rational points on varieties defined over a finite field. Mathematika, 28 (1981), 153–159.CrossRefGoogle Scholar
11.Schinzel, A., Schlickewei, H.-P. and Schmidt, W. M.. Small solutions of quadratic congruences and small fractional parts of quadratic forms. Ada Arith., 37 (1980), 241–248.CrossRefGoogle Scholar
12.Schlickewei, H.-P.. On indefinite diagonal forms in many variables. J. Reine Angew. Math., 307/8 (1979), 279–294.Google Scholar
13.Schmidt, W. M.. Equations over finite fields: an elementary approach, Lecture Notes in Math. 536 (Springer, Berlin, 1976).CrossRefGoogle Scholar
14.Schmidt, W. M.. Small fractional parts of polynomials (Amer. Math. Soc, Providence, 1977).Google Scholar
15.Schmidt, W. M.. Small zeros of additive forms in many variables. Trans. Amer. Math. Soc., 248 (1979), 121–133.CrossRefGoogle Scholar
16.Schmidt, W. M.. Bounds for exponential sums. To appear.Google Scholar
17.Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge University Press, Cambridge, 1981).Google Scholar