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

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