The Diophantine problem in a group (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 discuss the Diophantine problem in the groups the classical matrix groups Gn(R), where R is an associative unitary ring, n > 2, and Gn(R) is one of the groups GLn(R), SLn(R), Tn(R), UTn(R), PGLn(R), or PSLn(R) (in the last two cases we assume that R is also commutative). The main result is that the Diophantine problem in Gn(R) with a chosen set of coefficients is Ptime reducible (Karp reducible) to the Diophantine problem in R with respect to a suitable set of coefficients in R. What is much more interesting is that the converse is also true. In the case when the ring R is finitely generated and commutative the result above allows one to clarify the situation completely, modulo a big conjecture in number theory.
For not finitely generated rings, more so for uncountable rings, decidability of the Diophantine problem heavily depends on the set of constants. The case of classical fields of reals, complex and p-adic numbers is especially interesting, as well as the rings of p-adic integers (which is related to pro-p completions of groups). I am going to touch on this subject and, if time permits, show some surprising examples.
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
