• The density of rational points on non-singular hypersurfaces

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    • 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


      D R Heath-Brown1

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

  • Proceedings – Mathematical Sciences | News

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      Posted on July 25, 2019

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