哈里森中心与幂和乘积
Harrison center and products of sums of powers
摘要：
本文主要研究\[ (x_1^d + x_2^d + \cdots + x_r^d) (y_1^d + y_2^d + \cdots y_n^d) = z_1^d + z_2^d + \cdots + z_n^d \]等恒等式，其中$d>2,$、$x=(x_1, x_2, \dots, x_r)$和$y=(y_1, y_2, \dots, y_n)$是不定方程组，每个$z_k$是$y$中的线性形式，其系数在特征$0$或大于$d.$的任意域$\k$上的有理函数域$\k (x)$中。这些恒等式是著名的Hurwitz、Radon和Pfister平方和合成公式的高阶模拟。我们证明，至少$3$的度幂和的合成等式是微不足道的，即，如果$d>2,$则$r=1.$我们的证明是简单而初等的，其中的关键是Harrison的齐次多项式中心理论。
英文摘要：
This paper is mainly concerned with identities like \[ (x_1^d + x_2^d + \cdots + x_r^d) (y_1^d + y_2^d + \cdots y_n^d) = z_1^d + z_2^d + \cdots + z_n^d \] where $d>2,$ $x=(x_1, x_2, \dots, x_r)$ and $y=(y_1, y_2, \dots, y_n)$ are systems of indeterminates and each $z_k$ is a linear form in $y$ with coefficients in the rational function field $\k (x)$ over any field $\k$ of characteristic $0$ or greater than $d.$ These identities are higher degree analogue of the well-known composition formulas of sums of squares of Hurwitz, Radon and Pfister. We show that such composition identities of sums of powers of degree at least $3$ are trivial, i.e., if $d>2,$ then $r=1.$ Our proof is simple and elementary, in which the crux is Harrison's center theory of homogeneous polynomials.
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