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Representation of a rational number as a sum of ninth or higher odd powers. (English) Zbl 1440.11043
Summary: In the present paper, we substantially generalize one of the results obtained in our earlier paper [the author, Int. J. Number Theory 9, No. 4, 867–876 (2013; Zbl 1268.11049)]. We present a solution of a problem of Waring type: if \(F(x_1, \ldots ,x_N)\) is a symmetric form of odd degree \(n\geq 9\) in \(N=16\cdot 2^{n-9}\) variables, then for any \(q\in \mathbb{Q}\), \(q\neq 0\), the equation \(F(x_i)=q\) has rational parametric solutions, that depend on \(n-8\) parameters.
MSC:
11D72 Diophantine equations in many variables
11P05 Waring’s problem and variants
Citations:
Zbl 1268.11049
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References:
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