Quantifier Elimination and Invariant Theory: Applications to Quaternions, Octonions, and Other Algebras

TL;DR

This paper applies invariant theory and a general quantifier elimination method to classical algebras like quaternions and octonions, solving open questions and extending the scope of algebraic decision procedures.

Contribution

It introduces a novel approach combining invariant theory with quantifier elimination to analyze classical algebras, addressing previously unresolved problems.

Findings

Quantifier elimination results for quaternions and octonions.

Resolution of an open question in algebraic decision problems.

Extension of methods to various finite-dimensional algebras.

Abstract

We build on our previous paper \cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other classes of finite-dimensional algebras over real closed and algebraically closed fields. In particular, the first two examples answer an open question posed recently in \cite{savi}.