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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
Zbl 1268.11049
Full Text:
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