Inhomogeneous additive equations

Abstract

In this paper, we study the function [Formula: see text], which we define as the smallest number [Formula: see text] of variables needed to guarantee that the equation [Formula: see text] has nontrivial solutions in each of the [Formula: see text]-adic fields [Formula: see text], regardless of the rational integer coefficients. This generalizes the [Formula: see text] function of Davenport and Lewis. In this paper, we give a sharp upper bound for [Formula: see text] and compute its value for various choices of the degrees.