Closed Points on Cubic Hypersurfaces

Integral Points on Cubic Hypersurfaces

Let g ∈ Z[x1, . . . , xn] be an absolutely irreducible cubic polynomial whose homogeneous part is non-degenerate. The primary goal of this paper is to investigate the set of integer solutions to the equation g = 0. Specifically, we shall try to determine conditions on g under which we can show that there are infinitely many solutions. An obvious necessary condition for the existence of integer ...

Counting Rational Points on Cubic Hypersurfaces: Corrigendum

R0<b162R0 gcd(b1, N )1/2 R 0 (HP) . The second line is false and in fact one has M1 = 1 in Proposition 3. The author is very grateful to Professor Hongze Li for drawing his attention to this flaw. The error can be fixed by introducing an average over b1 into the statement of Proposition 3. This allows us to recover the main theorem in [1], and also [2, Lemma 11], via the following modification....

Counting Rational Points on Cubic Hypersurfaces

Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dimX − 4. Then the main result in this paper is a proof of the fact that X(Q) contains Oε,X(B ) points of height at most B.

N ov 2 00 6 INTEGRAL POINTS ON CUBIC HYPERSURFACES

RATIONAL POINTS ON CUBIC HYPERSURFACES OVER Fq(t)

The Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field Fq(t), provided that char(Fq) > 3 and X has dimension at least 6.