Harrison Center and Products of Sums of Powers

Abstract

This paper is mainly concerned with identities like

\begin{array}{c}\hfill \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{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left( x_1^d + x_2^d + \cdots + x_r^d\right) \left( y_1^d + y_2^d + \cdots y_n^d\right) = z_1^d + z_2^d + \cdots + z_m^d \end{aligned}$$\end{document}

where

d>2,

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d>2,$$\end{document}

m\in \{n,n+1\},

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \in \{n, n+1\},$$\end{document}

x=(x_1,x_2,\cdots ,x_r)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x=(x_1, x_2, \dots , x_r)$$\end{document}

and

y=(y_1,y_2,\cdots ,y_n)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$y=(y_1, y_2, \dots , y_n)$$\end{document}

are systems of indeterminates and each

z_k

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$z_k$$\end{document}

is a linear form in y with coefficients in the rational function field

\mathbb{k}(x)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {k}(x)$$\end{document}

over any field

\mathbb{k}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {k}$$\end{document}

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 / 11E76

References

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Funding

NSFC(11971181)

NSFC(12131015)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature