Existentially closed fields with G-derivations

We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.Comment: In version 2: new proof of (the current) Proposition 3.3

Integrating Hasse-Schmidt derivations

We study integrating (that is expanding to a Hasse-Schmidt derivation) derivations, and more generally truncated Hasse-Schmidt derivations, satisfying iterativity conditions given by formal group laws. Our results concern the cases of the additive and the multiplicative group laws. We generalize a theorem of Matsumura about integrating nilpotent derivations (such a generalization is implicit in work of Ziegler) and we also generalize a theorem of Tyc about integrating idempotent derivations

Undecidability in number theory

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342