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

Short bases of lattices over number fields

Abstract. Lattices over number elds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, short (pseudo)-basis is important in many applications. In this article, we provide the rst algorithm that computes such a short (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to nd a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a eld invariant. As it runs in polynomial time, this provides an e ective variant of Minkowski's second theorem for lattices over number elds.