Small zeros of additive forms in many variables

Schmidt, Wolfgang M.

doi:10.1090/S0002-9947-1979-0521696-3

Small zeros of additive forms in many variables
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by Wolfgang M. Schmidt PDF

Trans. Amer. Math. Soc. 248 (1979), 121-133 Request permission

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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