On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces

Mei Chu Chang, Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface
ƒ1(x1) +···+ ƒn(xn) = ax1k1 ...xnkn
for some polynomials ƒ i ∈ Z[] and nonzero integers a and ki, i = 1,...,n. In the case of 
ƒ1(X) = ··· = ƒn(X) = 2   and   k1 =··· = kn = 1
the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for ƒ1(X) = ··· = ƒn(X) = n   and   k1 =··· = kn = 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.
Original languageEnglish
Pages (from-to)935-954
Number of pages20
JournalMathematische Zeitschrift
Volume282
Issue number3-4
DOIs
Publication statusPublished - Apr 2016
Externally publishedYes

Keywords

  • Congruences
  • Integer points on hypersurfaces
  • Multiplicative character sums

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