Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T15:42:06.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Small Solutions of Congruences in a Large Number of Variables1

Published online by Cambridge University Press:  20 November 2018

Wolfgang M. Schmidt
Wolfgang M. Schmidt*
Affiliation:
University of Colorado Boulder, Colorado, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that a system of congruences 1(x) ≡ . . . ≡

(x) = 0 (mod m)

where each i(x) = i,(x1, .. . ,x2,) is a form of degree at most k has a nontrivial solution x satisfying |xi|≦cm(½)+∊ (i=1,...,S)

with c = c(k,r,∊), provided that ∊ > 0 and that S > S1(k,r,∊).

Keywords

10B3010C10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 295 - 305
Copyright
Copyright © Canadian Mathematical Society 1985

Footnotes

1

Written with partial support from NSF-MCS-8211461.

References

1. Baker, R.C., Small solutions of quadratic and quartic congruences, Mathematika 27 (1980), pp. 30–45.Google Scholar
2. Baker, R.C., Small solutions of congruences, Mathematika 30 (1983), pp. 164–188.Google Scholar
3. Baker, R.C. and Harman, G., Small fractional parts of quadratic and additive forms, Math. Proc. Camb. Phil. Soc. 90 (1981), pp. 5–12.Google Scholar
4. Birch, B., Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), pp. 102–105.Google Scholar
5. Brauer, R., A note on systems of homogeneous algebraic equations, Bull. A.M.S. 51 (1945), pp. 749–755.Google Scholar
6. Cassels, J.W.S., An introduction to diophantine approximation, Cambridge Tracts in Math, and Math. Physics, 45 (1957).Google Scholar
7. Schinzel, A., Schlickewei, H.P. and Schmidt, W.M., Small solutions of quadratic congruences and small fractional parts of quadratic forms, Acta Arith. 37 (1980), pp. 241—248 Google Scholar
8. Schmidt, W.M., Diophantine inequalities for forms of odd degree, Advances in Math. 38 (1980), pp. 128–151.Google Scholar
9. Schmidt, W.M., Bounds for exponential sums, Acta Arith. (to appear).Google Scholar