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. 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.
English (ENGL) | Penn State
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