T. D. Browning

most citedOn Manin's conjecture for singular del Pezzo surfaces of degree four, I

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math.NT2005★ 1 cited

Simultaneous equal sums of three powers

T. D. Browning, D. R. Heath-Brown

Using a recent result of Salberger, we establish the paucity of non-trivial positive integer solutions to a certain system of diagonal Diophantine equations.

math.NT2005

Equal sums of like polynomials

T. D. Browning

Let $f$ be a polynomial of degree $d > 6$ , with integer coefficients. Then the paucity of non-trivial positive integer solutions to the equation $f (a) + f (b) = f (c) + f (d)$ is established.…

math.NT2005

Plane curves in boxes and equal sums of two powers

T. D. Browning, D. R. Heath-Brown

Given an absolutely irreducible ternary form $F$ , the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are rest…

math.NT2005

The density of rational points on non-singular hypersurfaces, II

T. D. Browning, D. R. Heath-Brown

This paper establishes the conjecture that a non-singular projective hypersurface of dimension $r$ , which is not equal to a linear space, contains $O (B^{r + ε})$ rational points of h…

math.NT2005

The density of rational points on non-singular hypersurfaces, I

T. D. Browning, D. R. Heath-Brown

Let $X \subset P^{n}$ be a non-singular hypersurface of degree $d > 1$ , and let $ε > 0$ . This paper is concerned with the conjecture that there are $O (B^{n - 1 + ε})$ rational point…

math.NT2004★ 1 cited

On Manin's conjecture for singular del Pezzo surfaces of degree four, I

R. de la Breteche, T. D. Browning

In this paper the height zeta function associated to a certain singular del Pezzo surface of degree four is studied. If $U$ denotes the open subset formed by deleting the unique li…