Harrison center and products of sums of powers

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 $\text{k}(x)$ over any field $\text{k}$ of characteristic $0$ or greater than $d\mathrm{.}$ 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.