New solutions to the Hurwitz problem on square identities

TL;DR

This paper introduces a novel algebraic approach using $(Z_2)^n$-graded non-associative algebras to find new solutions to the classical Hurwitz problem on sum of squares identities, extending known identities.

Contribution

It develops a new algebraic framework that generalizes octonions, providing explicit formulas and discovering new solutions near the classical Hurwitz-Radon identity.

Findings

Explicit formula for Hurwitz-Radon identity

New solutions to the Hurwitz problem

Generalization of octonionic algebra

Abstract

The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbb{Z}_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical Hurwitz-Radon identity and leads to new solutions in a neighborhood of the Hurwitz-Radon identity.