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2 results

  • ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

  • Part of
  • IGOR E. SHPARLINSKI
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 89 / Issue 2 / April 2014
    Published online by Cambridge University Press:
    07 August 2013, pp. 300-307
    Print publication:
    April 2014
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  • We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

  • SOLUTIONS TO POLYNOMIAL CONGRUENCES IN WELL-SHAPED SETS

  • Part of
  • BRYCE KERR
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
    Published online by Cambridge University Press:
    12 June 2013, pp. 435-447
    Print publication:
    December 2013
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  • 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.