Rigorous Arithmetic in the Arakelov Divisor Class Group of a Number Field.
July 27th, 2010This post presents a poster of mine presented at the poster session of the 9th Algorithmic Number Theory Symphoisum.
Inequalities.
February 9th, 2010This post shows a diagram, listing a lot of inequalities and showing implications between them.
How to Compute the 5-adic Expansion of 1/2; or: Hensel’s Lemma and (Non-Analytic) Newton Iteration.
February 6th, 2010In this post, we consider the quest of computing the 5-adic expansion of 1/2. We begin with introducing p-adic integers and numbers, and discussing when certain polynomials with coefficients in the integers have zeroes in the p-adic integers. This question is closely related to Hensel’s lemma, which can be proven using an algebraic version of Newton’s iteration. We use this to compute approximations of rational numbers in the p-adics, and consider which p-adic numers have an eventually periodic expansion.
Finding Lattice Points, Finite Abelian Groups, and Explaining Algorithms.
January 29th, 2010We compare the tasks of finding points of a lattice, computing the structure of finite abelian groups and explaining algorithms. We show up relations between these three topics and, as an example, depict the baby-step giant-step algorithm for order computation, as well as Terr’s modification of this algorithm.
Homomorphisms, Tensor Products and Certain Canonical Maps.
January 29th, 2010A standard topic in linear algebra is the dual space of a vector space, as well as the canonical embedding of a vector space in its double dual. Moreover, transposition of homomorphisms in terms of dual spaces is rather well known. Something less known is that one has a canonical map from the dual of V tensored with W to the space of homomorphisms from V to W. In this abstract nonsense post, we describe these canonical maps, their interplay, and try to determine their images.
Diagonalizable Matrices.
January 29th, 2010We consider the property of an n times n matrix of being diagonalizable. Is this property open in the standard topology, or the Zariski topology? The emphasis lies on the real and complex numbers, as well as on arbitrary algebraically closed fields.
Euclidean Domains, and the Extended Euclidean Algorithm.
November 18th, 2009We will discuss Euclidean domains together with a constructive proof of the fact that every two elements have a greatest common divisor, which is essentially the Euclidean algorithm.
We will state several (more or less) useful properties of the Extended Euclidean Algorithm, in particular for the case of integers and univariate polynomials over a field.
The Hasse derivative, part II: Multivariate partial Hasse derivatives.
October 2nd, 2009Following a suggestion by A. Maevskiy, we show how the Hasse derivative can be extended to partial Hasse derivative in arbitrary multivariate polynomial rings. We show multivariate versions of Taylor’s Formula, of the Identity Theorem, and of the Generalized Leibnitz Rule.
Fun With Representable Functors, or Why I Like Yondea’s Lemma.
August 16th, 2009We discuss the notion of representable functors in Category Theory. Then, we present Yoneda’s lemma and apply it to the situation of group objects in categories and their relation to functors into the category of groups, resulting in a surprising result that these two concepts are essentially the same. Most proofs are included, as well as lots of commutative diagrams.
About Base Changes and Tensor Products.
August 15th, 2009In Linear Algebra, one often has the problem that one wants to talk about complex eigenvalues of objects defined over the reals. If the object is a matrix, it is clear what that means. But what if the object is an endomorphism of a non-canonical real vectorspace? This question is strongly related an important use of tensor products, namely base changes.
