In 1970, Yuri Matiyasevich completed work begun by Martin Davis, Hilary Putnam, and Julia Robinson on Hilbert's 10th Problem. Hilbert's 10th Problem asks for a universal algorithm to decide whether a polynomial equation with integer coefficients has a solution. The result of Davis, Putnam, Robinson, and Matiyasevich showed that no such algorithm exists, but gave in fact the much stronger result that "all recursively enumerable sets are diophantine".
Since then, much work has been done extending the unsolvability of Hilbert's 10th Problem to other rings, and somewhat less on the problem of carrying over the stronger result to other rings. This poster presents a survey of results of the latter type, paying particular attention to recent work on the polynomial rings over finite fields.