Let n 4, and let Q âˆˆ Z[X1,. .. , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Qâ€¦ (More)

For any n > 3, let F âˆˆ Z[X0, . . . , Xn] be a form of degree d > 5 that defines a non-singular hypersurface X âŠ‚ P. The main result in this paper is a proof of the fact that the number N(F ; B) ofâ€¦ (More)

Let n > 4, and let Q âˆˆ Z[X1, . . . , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and whenâ€¦ (More)

This paper contains a proof of the Manin conjecture for the singular del Pezzo surface X : x0x1 âˆ’ x2 = x0x4 âˆ’ x1x2 + x3 = 0, of degree four. In fact, if U âŠ‚ X is the open subset formed by deletingâ€¦ (More)

For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauerâ€“Manin obstruction controls weak approximation. The proof is based onâ€¦ (More)

For an irreducible polynomial in at most two variables the problem of representing power-free integers is investigated. Mathematics Subject Classification (2000). 11N32.

Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can beâ€¦ (More)

For any n â‰¥ 2, let F âˆˆ Z[x1, . . . , xn] be a form of degree d â‰¥ 2, which produces a geometrically irreducible hypersurface in Pnâˆ’1. This paper is concerned with the number N(F ;B) of rational pointsâ€¦ (More)

For given B â‰¥ 1 and Îµ > 0, we show that the number of rational points on a non-singular cubic surface, not lying on any line, and of height at most B, is OÎµ(B ) whenever the surface contains aâ€¦ (More)