An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

We give an algorithm to compute the following cohomology groups on U = \C^n \setminus V(f) for any non-zero polynomial f \in \Q[x_1, ..., x_n]; 1. H^k(U, \C_U), \C_U is the constant sheaf on $U$ with stalk \C. 2. H^k(U, \Vsc), \Vsc is a locally constant sheaf of rank 1 on $U$. We also give partial results on computation of cohomology groups on $U$ for a locally constant sheaf of general rank and on computation of H^k(\C^n \setminus Z, \C) where $Z$ is a general algebraic set. Our algorithm is based on computations of Gr\"obner bases in the ring of differential operators with polynomial coefficients.Comment: 38 page

Integral D-Finite Functions

We propose a differential analog of the notion of integral closure of algebraic function fields. We present an algorithm for computing the integral closure of the algebra defined by a linear differential operator. Our algorithm is a direct analog of van Hoeij's algorithm for computing integral bases of algebraic function fields

Computing generators of the unit group of an integral abelian group ring

We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalizable algebraic group

Certified lattice reduction

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants-where the rational arithmetic required by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.Comment: 23 page

Computing Invariants of Simplicial Manifolds

This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure