Orbit-finite systems of inequalities

A system of inequalities is orbit-finite if it is finite up to certain permutations of variables. In this talk, I will describe this concept using interesting examples, and present our recent results on the solvability of these systems. In particular, we have proven that the existence of finitely supported solutions of orbit-finite systems of inequalities is undecidable for integer solutions and decidable for rational solutions. This is a joint work with Piotr Hofman and Sławomir Lasota.