The Diophantine problem in commutative rings and solvable groups

Abstract: The Diophantine problem in a group or ring $G$ is decidable if there exists an algorithm that given a finite system of equations with coefficients in $G$ decides whether or not the system has a solution in $G$. I will overview recent developments that have been made in regards to this problem in the area of commutative rings and solvable groups. For large classes of such rings and groups the situation is completely clarified modulo a big conjecture in number theory. This includes the class of all finitely generated commutative rings (with or without unit), all finitely generated nilpotent groups, several polycyclic groups, and several matrix groups. The talk is based on joint results with Alexei Miasnikov and Denis Ovchinnikov .

group theoryrings and algebras

Audience: researchers in the topic

Al@Bicocca take-away