On the first-order theory of quaternions

Abstract

In complex and real algebraic geometry, the question on how projections of algebraic sets behave is completely understood by the celebrated Chevalley's theorem and Tarski-Seidenberg principle, respectively. In general, the latter geometric problem has a very clear interpretation in model theory, which in turn is also very useful for geometric applications. The aim of this talk is to investigate the quaternionic version of the latter problem, starting from a geometric point of view and developing a model theoretical approach. Then, the main results will concern fundamental properties of the first-order theory of quaternions such as completeness and model completeness with application on the study of slice regular polynomials and their zero loci.