Hilbert’s tenth problem asks whether there is an algorithm that given a polynomial with integer coefficients (in an arbitrary number of variables) determines whether it has integer roots. A celebrated theorem of Davis, Matiyasevich, Putnam and Robinson shows that this is not the case. Their result shows that the class of *Diophantine* sets coincides with the *a priori* larger class of r.e. (or ) sets.

We proved this result under the assumption that exponentiation is Diophantine. This is the key result, and will be dealt with in the following lecture.

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