The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proo...

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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 var...

In this paper, an upper bound for the number of integral points of bounded height on an affine complete intersection defined over is proven. The proof uses an extension to complete intersections of the method used for hypersurfaces by Heath-Brown (Th...

We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, where the $f_i$ are polynomials with integer coefficients. The estimates are obtained by generalising a...

We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the determinant method developed by Heath-Brown, along with recent results by Salber...

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proo...

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 var...

We study the hexagonal lattice $\mathbb{Z}[\omega]$, where $\omega^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on average, and sug...

Let f be a polynomial of degree at least four with integer-valued coefficients. We establish new bounds for the density of integer solutions to the equation f=0, using an iterated version of Heath-Browns q-analogue of van der Corput's method of expon...