WARING ’ S PROBLEM FOR POLYNOMIALS Stephen McAdam

SECTION 1: INTRODUCTION. Lagrange proved that any positive integer was the sum of four or fewer numbers of the form x with x a positive integer. Waring asked if given an n ≥ 2, there is an f = f(n) such that every positive integer is the sum of f or fewer numbers of the form x with x a positive integer. Hilbert showed the answer was yes, via a very difficult and sophisticated proof. Subsequently, Y. V. Linnik discovered an elementary proof, reported in chapter 3 of the lovely little book Three pearls of Number Theory by A. Y. Khinchin, [K], (at this writing, available from Dover Press). We here present a rewriting of that chapter, and also carry Linnik’s ideas somewhat further. In particular, corollary 3 below will show that if P(X) is a non-constant polynomial with integral coefficients and with positive leading coefficient, and if there is an integer z with P(z) = 1, then there is an f such that all positive integers are the sum of f or fewer numbers of the form P(x) with P(x) > 0. Waring’s problem concerns the special case P(X) = X, for which P(1) = 1.