• The density of rational points on non-singular hypersurfaces

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/104/01/0013-0029

• # Keywords

Rational points; hypersurface; counting function; multiple exponential sum; Deligne’s bounds; singular locus

• # Abstract

LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤn;F(x)=0, |x|⩽X}, where$$\left| x \right| = \mathop {max}\limits_{1 \leqslant r \leqslant n} \left| {x_r } \right|$$. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪Xn−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.

• # Author Affiliations

1. Magdalen College, Oxford - OXI 4AU, England

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019