Abstract

Abstract The methods of this chapter are mostly elementary number theory. We want to count the number of integer solutions in bounded regions for certain sets of equations and inequalities. The difficulties are partly algebraic, partly analytic. A system of polynomial equations defines an algebraic variety in affine space. The geometry is dominated by two integers, the dimension and the genus of the variety. The size of the set of integer solutions is usually much smaller than the dimension of the variety would suggest. In the case of projective curves, there are only finitely many integer solutions unless the genus is zero or one (Fallings 1983) (an integer point on a projective curve corresponds to a rational point on an affine curve).

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