On the distribution of points on the generalized markoff-hurwitz and dwork hypersurfaces

Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface f1(x1) fn(xn) ≡ x 1{k1} xn kn(mod p)for some polynomials fi [X] that are not constant modulo a prime p and integers ki with gcd(ki, p-1) = 1, i = 1, n. In the case of f1(X) fn(X) = aX2 and k1 kn =1 the above congruence is known as the Markoff-Hurwitz hypersurface, while for f1(X)=fn(X) = Xn and k1=kn =1 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 p1/2, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

Original languageEnglish
Pages (from-to)151-160
Number of pages10
JournalInternational Journal of Number Theory
Volume10
Issue number1
DOIs
Publication statusPublished - Feb 2014

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