In mathematics,
diophantine geometry is one approach to the theory of
Diophantine equations, formulating questions about such equations in terms of
algebraic geometry over a
ground field K that is not
algebraically closed, such as the field of
rational numbers or a
finite field, or more general
commutative ring such as the integers. A single equation defines a
hypersurface, and simultaneous Diophantine equations give rise to a general
algebraic variety V over
K; the typical question is about the nature of the set
V(
K) of points on
V with co-ordinates in
K, and by means of
height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number. Given the geometric approach, the consideration of
homogeneous equations and
homogeneous co-ordinates is fundamental, for the same reasons that
projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e.
lattice points) can be treated in the same way as an
affine variety may be considered inside a projective variety that has extra
points at infinity.