An Analogue to Minkowski's Geometry of Numbers in a Field of Series

Abstract

The inequality F(X) ? 1 defines a convex body in Rn which has its centre at the origin X = 0. Suppose that this body has the volume V. The well known result of Minkowski asserts that if V _ 2 , then the body contains at least one (and so at least two) lattice points different from 0. This theorem is contained in the following deeper result of Minkowski (G.d.Z. ??50-53): There are n independent lattice points X), X(2) ... , X An) in Rn with the following properties: (1) F(X(1') = arl) is the minimum of F(X) in all lattice points X $ 0, and for k _ 2, F(X(k)) = a(k) is the minimum of F(X) in all lattice points X which are independent of V), (k-1) (2) The determinant D of the points X, ..X(n) satisfies the inequalities