On linear equations over split-octonions

TL;DR

This paper characterizes the solution sets of linear equations over split-octonions, describing their structure, dimensions, and properties of solutions, including invertibility conditions, over algebraically closed fields.

Contribution

It provides a detailed geometric and algebraic analysis of linear equations over split-octonions, including solution variety descriptions and invertibility criteria.

Findings

Solution varieties are explicitly described as affine varieties.

Dimensions of solution sets are determined for arbitrary linear monomial equations.

Nonzero constant term equations with multiple solutions have invertible solutions.

Abstract

Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations $a(xb)=c$ and $a(bx)=c$ over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial equations in the split-octonions. Moreover, we show that if a linear monomial equation over the split-octonions with nonzero constant term has at least two solutions, then it necessarily possesses an invertible solution.