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Small zeros of additive forms in several variables

Authors:

J. S. Hwang

Affiliation:

(1) Institute of Mathematics, Academia Sinica, 11529 Taipei, Taiwan China

Abstract:

Letf(X) be an additive form defined by

$f(X) = f(x_1 ,x_2 , cdots ,x_3 ) = sigma _1 a_1 x_{_1 }^k + sigma _2 a_2 x_2^k + cdots sigma _3 a_3 x_{_3 }^k ,$

wherea _i≠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,x ₁,x ₂, …,x ₃ 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

Keywords:

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