SIMULTANEOUS DIAGONAL EQUATIONS OVER CERTAIN P-ADIC FIELDS

Let O be the valuation ring of a p-adic field K. Given a system S = (S1, ..., Sr) of diagonal forms of degree d > 1 in n variables over O, one obtains a map F: Or → R {smile} {∞} by assigning to each a ∈ Or the "solution density" of S(x) = a. The objective of the present paper is to study some properties of F and its Fourier transform, and as a by-product we shall obtain the result that S has a common non-trivial zero in Kn, provided that n > 2dr, d is a unit in O, and the residue field of O contains sufficiently many elements. © 1990.