On Projections of Free Semialgebraic Sets

TL;DR

This paper explores the possibility of a projection theorem in non-commutative real algebraic geometry, demonstrating limitations and establishing a weak version for certain types of projections.

Contribution

It reviews the challenges of defining free semialgebraic sets and proves a weak projection theorem for linear and separated variables in the non-commutative setting.

Findings

Full free projection theorem is unlikely due to undecidability issues.

A weak projection theorem holds for linear and separated variables.

The notion of free semialgebraic sets remains an open question.

Abstract

An important result in real algebraic geometry is the projection theorem: every projection of a semialgebraic set is again semialgebraic. This theorem and some of its conclusions lie at the basis of many other results, for example the decidability of the theory of real closed fields, and almost all Positivstellens\"atze. Recently, non-commutative real algebraic geometry has evolved as an exciting new area of research, with many important applications. In this paper we examine to which extend a projection theorem is possible in the non-commutative (=free) setting. Although it is not yet clear what the correct notion of a free semialgebraic set is, we review and extend some results that count against a full free projection theorem. For example, it is undecidable whether a free statement holds for all matrices of at least one size. We then prove a weak version of the projection theorem: projections along linear and separated variables yields a semi-algebraically parametrised free semi-algebraic set.