  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.
