Hua’s Lemma and Exponential Sums over Binary Forms

Abstract

AbstractWe establish mean value estimates for exponential sums over binary forms of strength comparable with the bounds attainable via classical, single variable estimates for diagonal forms. These new mean value estimates strengthen earlier bounds of the author when the degree d of the satisfies 5≤d≤ 10,, the improvements stemming from a basic lemma which provides uniform estimates for the number of integral points on affine plane curves in mean square. Exploited by means of the Hardy-Littlewood method, these estimates permit one to establish asymptotic formulae for the number of integral zeros of equations defined as sums of binary forms of the same degreedprovided that the number of variables exceeds\( \frac{{17}} {{16}}2d, \) improving significantly on what is attainable either by classical additive methods, or indeed the general methods of Birch and Schmidt.