Plane curves in boxes and equal sums of two powers

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 restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers.Comment: 15 pages; to appear in Math. Zei

Quadratic polynomials represented by norm forms

The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the rationals containing the roots of P. The proof uses analytic methods.Comment: 55 page

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

For any integers $d,n \geq 2$, let $X \subset \mathbb{P}^{n}$ be a non-singular hypersurface of degree $d$ that is defined over $\mathbb{Q}$. The main result in this paper is a proof that the number $N_X(B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N_X(B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon>0$. The implied constant in this estimate depends at most upon $d, \varepsilon$ and $n$

Integral points on cubic hypersurfaces

Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k. We show that the equation g=0 has infinitely many integer solutions when the cubic part of g defines a projective hypersurface with singular locus of dimension <n-10. The proof is based on the Hardy-Littlewood circle method.Comment: 18 page

Counting rational points on quadric surfaces

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.Comment: 29 page