Small zeros of additive forms in several variables

Abstract

Letf(X) be an additive form defined by $$f(X) = f(x_1 ,x_2 , \cdots ,x_3 ) = \sigma _1 a_1 x_{_1 }^k + \sigma _2 a_2 x_2^k + \cdots \sigma _3 a_3 x_{_3 }^k ,$$ whereai≠0 is integer,i=1,2…,s. In 1979, Schmidt proved that if ∈>0 then there is a large constantC(k,∈) such that fors>C(k,∈) the equationf(X)=0 has a nontrivial, integer solution in σ1, σ2, …, σ3,x1,x2, …,x3 satisfying $$\sigma _i = \pm and \left| {x_1 } \right| \leqslant A^e , i = 1,2, \cdots ,s where A = \mathop {max}\limits_{1 \leqslant i \leqslant s} \left| {a_1 } \right|.$$ Schmidt did not estimate this constantC(k,∈) since it would be extremely large. In this paper, we prove the following result