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)≪X n−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.
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Dedicated to the memory of Professor K G Ramanathan
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Heath-Brown, D.R. The density of rational points on non-singular hypersurfaces. Proc Math Sci 104, 13–29 (1994). https://doi.org/10.1007/BF02830871
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DOI: https://doi.org/10.1007/BF02830871