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摘要翻译：
利用两个本原解，给出了一个6变量的5次齐次任意对称diophantine方程的参数解。然后通过证明以下结果，我们将这种方法推广到任意奇次的对称形式。(1)$6\cdot2^{n-5}$变量中奇数次$n\ge5$的每一对称形式都有依赖于$2n-8$参数的有理参数解。（2）设$f(x_1,...,x_N)$为$n=6\cdot2^{n-4}$变量中奇数次$n\ge5$的对称形式，设$q$为任意有理数。那么方程$f(x_i)=q$有一个有理参数解，它依赖于$2n-6$参数。后一个结果可以看作是这类形式的一个Waring型问题的解决。
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英文标题：
《Symmetric homogeneous diophantine equations of odd degree》
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作者：
M. A. Reynya
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degree $n\ge 5$ in $6 \cdot 2^{n-5}$ variables has a rational parametric solution depending on $2n-8$ parameters. (2) Let $F(x_1, ..., x_N)$ be a symmetric form of odd degree $n\ge 5$ in $N=6 \cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the equation $F(x_i)=q$ has a rational parametric solution depending on $2n-6$ parameters. The latter result can be viewed as a solution of a problem of Waring type for this class of forms.
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PDF链接：
https://arxiv.org/pdf/0809.3973
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