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

Chang, Mei Chu; Shparlinski, Igor E.

doi:10.1007/s00209-015-1571-z

Cited by 1 publication

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“…Nontrivial upper bounds for S k (f ; N, H), particularly when H i is "short" relative to q i , are expected to have a variety of applications, for example to counting integral points on certain hypersurfaces, such as multi-dimensional generalizations of the Markoff-Hurwitz and Dwork hypersurfaces (see related work [16], [9]). We will prove bounds that are nontrivial when H i ≫ q 1/4+ε i by developing a multidimensional version of the Burgess method that allows us to apply recent results of Parsell, Prendiville and Wooley [14] on multi-dimensional Vinogradov Mean Value Theorems.…”

Section: Introductionmentioning

confidence: 99%

Burgess bounds for multi-dimensional short mixed character sums

Pierce

2016

Journal of Number Theory

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Abstract. This paper proves Burgess bounds for short mixed character sums in multidimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f , and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in k ≥ 1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation-dilation invariant system constructed in terms of f , we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions k ≥ 2.

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Section: Introductionmentioning

confidence: 99%