Harrison Center and Products of Sums of Powers

Abstract

This paper is mainly concerned with identities like

$\begin{aligned} \left( x_1^d + x_2^d + \cdots + x_r^d\right) \left( y_1^d + y_2^d + \cdots y_n^d\right) = z_1^d + z_2^d + \cdots + z_m^d \end{aligned}$

where $d>2,$ $m \in \{n, n+1\},$ $x=(x_1, x_2, \dots , x_r)$ 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 $\mathbb {k}(x)$ over any field $\mathbb {k}$ 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 all but one such composition identities of sums of higher powers are trivial, i.e., r necessarily equals 1. Our proof is simple and elementary, in which the crux is Harrison’s center theory of homogeneous polynomials.

Keywords

Sum of powers / Composition

Funding

NSFC(11971181)

NSFC(12131015)