Counting solutions to Diophantine equations

Doctoral thesis, 2010

This thesis presents various results concerning the density of rational and integral points on algebraic varieties. These results are proven with methods from analytic number theory as well as algebraic geometry. Using exponential sums in several variables over finite fields, we prove upper bounds for the number of integral points of bounded height on an affine variety. More precisely, our method is a generalization of a technique due to Heath-Brown — a multi-dimensional version of van der Corput’s AB-process. It yields new estimates for complete intersections of r hypersurfaces of degree at least three in A n , as well as for hypersurfaces in A n of degree at least four. We also study the so called determinant method, introduced by Bombieri and Pila to count integral points on curves. We show how their approach may be extended to higher-dimensional varieties to yield an alternative proof of Heath-Brown’s Theorem 14, avoiding p-adic considerations. Moreover, we use the determinant method to study the number of representations of integers by diagonal forms in four variables. Heath-Brown recently developed a new variant of the determinant method, adapted to counting points near algebraic varieties. Extending his ideas, we prove new upper bounds for the number of representations of an integer by a diagonal form in four variables of degree k ≥ 8. Furthermore, we use a refined version of the determinant method for affine surfaces, due to Salberger, to derive new estimates for the number of representations of a positive integer as a sum of four k-th powers of positive integers, improving upon estimates by Wisdom.