Harrison center and products of sums of powers

TL;DR

This paper investigates identities involving products of sums of powers and demonstrates that non-trivial identities of degree greater than two are essentially impossible, using Harrison's center theory of homogeneous polynomials.

Contribution

It proves that higher degree sum-of-powers identities are trivial, extending the understanding of composition formulas for sums of powers.

Findings

Identifies that non-trivial identities exist only for linear sums when degree > 2

Uses Harrison's center theory to provide a simple, elementary proof

Shows that higher degree sum-of-powers identities are trivial

Abstract

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.