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

Mei Chu Chang; Igor E. Shparlinski

doi:10.1007/s00209-015-1571-z

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 journal › Article › peer-review

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) +···+ ƒ_n(x_n) = ax₁^k1...x_n^kn
for some polynomials ƒ _i ∈ Z[X ] and nonzero integers a and k_i, i = 1,...,n. In the case of
ƒ₁(X) = ··· = ƒ_n(X) = X ² and k₁ =··· = k_n = 1
the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for ƒ₁(X) = ··· = ƒ_n(X) = X ⁿ and k₁ =··· = k_n = 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.

Original language	English
Pages (from-to)	935-954
Number of pages	20
Journal	Mathematische Zeitschrift
Volume	282
Issue number	3-4
DOIs	https://doi.org/10.1007/s00209-015-1571-z
Publication status	Published - Apr 2016
Externally published	Yes

Keywords

Congruences
Integer points on hypersurfaces
Multiplicative character sums

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title = "On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces",

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 ...xnknfor some polynomials ƒ i ∈ Z[X ] and nonzero integers a and ki, i = 1,...,n. In the case of ƒ1(X) = ··· = ƒn(X) = X 2 and k1 =··· = kn = 1the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for ƒ1(X) = ··· = ƒn(X) = X n and k1 =··· = kn = 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.",

keywords = "Congruences, Integer points on hypersurfaces, Multiplicative character sums",

author = "Chang, {Mei Chu} and Shparlinski, {Igor E.}",

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T1 - On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces

AU - Chang, Mei Chu

AU - Shparlinski, Igor E.

PY - 2016/4

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N2 - 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 ...xnknfor some polynomials ƒ i ∈ Z[X ] and nonzero integers a and ki, i = 1,...,n. In the case of ƒ1(X) = ··· = ƒn(X) = X 2 and k1 =··· = kn = 1the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for ƒ1(X) = ··· = ƒn(X) = X n and k1 =··· = kn = 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.

AB - 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 ...xnknfor some polynomials ƒ i ∈ Z[X ] and nonzero integers a and ki, i = 1,...,n. In the case of ƒ1(X) = ··· = ƒn(X) = X 2 and k1 =··· = kn = 1the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for ƒ1(X) = ··· = ƒn(X) = X n and k1 =··· = kn = 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.

KW - Congruences

KW - Integer points on hypersurfaces

KW - Multiplicative character sums

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ER -

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

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Keywords

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