Geometry Of Numbers

Abstract Introduction and restatement of the fundamental theorem. This chapter is an introduction to the ‘geometry of numbers’, the subject created by Minkowski on the basis of his fundamental Theorem 37 and its generalization in space of ndimensions We shall need the n-dimensional generalizations of the notions which we used in §§ 3.9–11; but these, as we said in § 3.11, are straightforward. We define a lattice, and equivalence of lattices, as in § 3.5, parallelograms being replaced by n-dimensional parallelepipeds; and a convex region as in the first definition of § 3.9.† Minkowski’s theorem is then Theorem 446. Any convex region in n-dimensional space, symmetrical about the origin and of volume greater than2n, contains a point with integral coordinates, not all zero. Any of the proofs of Theorem 37 in Ch. III may be adapted to prove Theorem 446: we take, for example, Mordell’s.