Harrison Center and Products of Sums of Powers

Hua-Lin Huang , Lili Liao , Huajun Lu , Yu Ye , Chi Zhang

Communications in Mathematics and Statistics ›› : 1 -8.

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Communications in Mathematics and Statistics ›› : 1 -8. DOI: 10.1007/s40304-023-00367-1
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Harrison Center and Products of Sums of Powers

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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.

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Sum of powers / Composition

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Hua-Lin Huang, Lili Liao, Huajun Lu, Yu Ye, Chi Zhang. Harrison Center and Products of Sums of Powers. Communications in Mathematics and Statistics 1-8 DOI:10.1007/s40304-023-00367-1

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Funding

NSFC(11971181)

NSFC(12131015)

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