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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small zeros of additive forms in many variables

Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 248 (1979), 121-133
MSC: Primary 10B30; Secondary 10J10
MathSciNet review: 521696
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Abstract: It is shown that if s is large as a function of k and of $ \varepsilon > 0$, then the diophantine equation $ {a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k$ with positive coefficients $ {a_1}, \ldots ,{a_s}$, $ {b_1}, \ldots ,{b_s}$ has a nontrivial solution in nonnegative integers $ {x_1}, \ldots ,{x_s}$, $ {y_1}, \ldots ,{y_s}$ not exceeding $ {m^{\left( {1/k} \right) + \varepsilon }}$, where m is the maximum of the coefficients.

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