Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T06:19:19.981Z Has data issue: false hasContentIssue false
  • English
  • Français

SOLUTIONS TO POLYNOMIAL CONGRUENCES IN WELL-SHAPED SETS

Part of: Diophantine equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  12 June 2013

BRYCE KERR
BRYCE KERR*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia email bryce.kerr@students.mq.edu.au
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.

We use a generalisation of Vinogradov’s mean value theorem of Parsell et al. [‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331] and ideas of Schmidt [‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396] to give nontrivial bounds for the number of solutions to polynomial congruences, when the solutions lie in a very general class of sets, including all convex sets.

Keywords

polynomial congruencesdistribution of points

MSC classification

Secondary: 11D79: Congruences in many variables 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 435 - 447
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bourgain, J., Garaev, M. Z., Konyagin, S. V. and Shparlinski, I. E., ‘On the hidden shifted power problem’, SIAM J. Comput. 41 (2012), 1524–1557.Google Scholar
Bourgain, J., Garaev, M. Z., Konyagin, S. V. and Shparlinski, I. E., ‘On congruences with products of variables from short intervals and applications’, Proc. Steklov Inst. Math. 280 (2013), 67–96.Google Scholar
Chang, M.-C., Cilleruelo, J., Garaev, M. Z., Hernández, J., Shparlinski, I. E. and Zumalacárregui, A., ‘Concentration of points and isomorphism classes of hyperelliptic curves over a finite field in some thin families’, arXiv:1111.1543.Google Scholar
Cilleruelo, J., Garaev, M., Ostafe, A. and Shparlinski, I. E., ‘On the concentration of points of polynomial maps and applications’, Math. Z. 272 (2012), 825–837.Google Scholar
Fouvry, É., ‘Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums’, Israel J. Math. 120 (2000), 81–96.Google Scholar
Fouvry, É. and Katz, N., ‘A general stratification theorem for exponential sums, and applications’, J. reine angew. Math. 540 (2001), 115–166.Google Scholar
Granville, A., Shparlinski, I. E. and Zaharescu, A., ‘On the distribution of rational functions along a curve over ${ \mathbb{F} }_{p} $ and residue races’, J. Number Theory 112 (2005), 216–237.CrossRefGoogle Scholar
Luo, W., ‘Rational points on complete intersections over ${ \mathbb{F} }_{p} $’, Int. Math. Res. Not. IMRN 1999 (1999), 901–907.CrossRefGoogle Scholar
Parsell, S., Prendiville, S. and Wooley, T., ‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331.Google Scholar
Schmidt, W., ‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396.Google Scholar
Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 44 (Cambridge University Press, 1993).Google Scholar
Shparlinski, I. E., ‘On the distribution of points on multidimensional modular hyperbolas’, Proc. Japan Acad. Sci. Ser. A 83 (2007), 5–9.Google Scholar
Shparlinski, I. E., ‘On the distribution of solutions to polynomial congruences’, Archiv. Math. 99 (2012), 345–351.Google Scholar
Shparlinski, I. E. and Skorobogatov, A. N., ‘Exponential sums and rational points on complete intersections’, Mathematika 37 (1990), 201–208.CrossRefGoogle Scholar
Skorobogatov, A. N., ‘Exponential sums, the geometry of hyperplane sections, and some Diophantine problems’, Israel J. Math. 80 (1992), 359–379.Google Scholar
Weyl, H., ‘On the volume of tubes’, Amer. J. Math. 61 (1939), 461–472.Google Scholar
Zumalacárregui, A., ‘Concentration of points on modular quadratic forms’, Int. J. Number Theory 7 (2011), 1835–1839.Google Scholar