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Journals and Conferences
Let n 4, and let Q ∈ 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 Q… (More)
For any n > 3, let F ∈ Z[X0, . . . , Xn] be a form of degree d > 5 that defines a non-singular hypersurface X ⊂ P. The main result in this paper is a proof of the fact that the number N(F ; B) of… (More)
Let n > 4, and let Q ∈ 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… (More)
This paper contains a proof of the Manin conjecture for the singular del Pezzo surface X : x0x1 − x2 = x0x4 − x1x2 + x3 = 0, of degree four. In fact, if U ⊂ X is the open subset formed by deleting… (More)
For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on… (More)
For an irreducible polynomial in at most two variables the problem of representing power-free integers is investigated. Mathematics Subject Classification (2000). 11N32.
We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).
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 be… (More)
For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in Pn−1. This paper is concerned with the number N(F ;B) of rational points… (More)
For given B ≥ 1 and ε > 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 Oε(B ) whenever the surface contains a… (More)