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ON THE DISTRIBUTION OF POINTS ON THE GENERALIZED MARKOFF–HURWITZ AND DWORK HYPERSURFACES

IGOR E. SHPARLINSKI

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

https://doi.org/10.1142/S1793042113500863Cited by:2 (Source: Crossref)

Abstract

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface

for some polynomials f_i ∈ ℤ[X] that are not constant modulo a prime p and integers k_i with gcd(k_i, p-1) = 1, i = 1, …, n. In the case of

the above congruence is known as the Markoff–Hurwitz hypersurface, while for

it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution 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.

Keywords:

AMSC: 11D79, 11K38, 11L07, 11L40

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