We show how Partial Fraction Decomposition is a consequence of the Chinese Remainder Theorem. Our results hold for arbitrary principal ideal domains, with stronger results for certain Euclidean rings. Therefore, we do not only have Partial Fraction Decomposition for rational function fields, but also for the rational numbers.
In this post, we derive an explicit lower bound for the number of pairs of coprime integers bounded by some n.
This short post discusses a cute statement which shows the power of a certain subset of axioms of a ring, including prominently the distributive law.
Today I present a cute identity which appeared while explicitly computing the Gram-Schmidt orthogonalization of a base.
We present a (well-known) method to compute a solution to the linear system Ax=b over the integers, when it is known that the determinant of A is non-zero and that a solution with integral coefficients exists. We also provide a running time analysis.
In this post, I show how to explicitly compute a determinant. This determinant allows me to write down a closest solution in the 2-norm to a certain unsolvable linear system.
We show how to prove a number theoretic inequality, originating from geometry, using an elementary approach.
This post shows a way to quickly show that the determinant is multiplicative without getting your hands dirty.
This post presents a poster of mine presented at the poster session of the 9th Algorithmic Number Theory Symphoisum.
This post shows a diagram, listing a lot of inequalities and showing implications between them.
In 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.
We 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.
A 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.
We 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.
We 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.
Following 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.
We 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.
In 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.
Functional Calculus in Linear Algebra, the Jordan Decomposition Reloaded and Cayley-Hamilton's Theorem.
We explain the aims of functional calculus and specialize to polynomials evaluated at endomorphisms. We reconsider the Jordan decomposition and prove it with more generality. Then, we discuss Taylor expansion in the nilpotent part for endomorphisms with separable minimal polynomials, and prove Cayley-Hamilton again for arbitrary fields.
In real and complex analysis, the Taylor series expansion is a very important tool. For polynomials over arbitrary unitary rings, it is possible to define a derivative which behaves similar to the usual derivative; unfortunately, the Identity Theorem and Taylor's formula do not transfer to this new situation. Fortunately, there exists a different definition of derivatives for these cases, namely the Hasse derivative. Not only does it gives a Identity Theorem and Taylor's formula back, but also allows to write other identities in a simpler way.
We show how to obtain n-dimensional infrastructures from global fields of unit rank n. We will also discuss how to obtain baby steps in these cases, and show graphical representations of certain two-dimensional infrastructures obtained from function fields.
We explain a general technique to obtain a reduction map, given X and d and, varying with the method of construction, additional information for every x in X. Moreover, we explain a technique on how to obtain n-dimensional infrastructures from certain lattices in (n+1)-dimensional space.
We will introduce n-dimensional infrastructures and briefly discuss reductions, f-representations and giant steps. We will also discuss how infrastructures can be obtained from finite abelian groups.
We introduce the notion of f-representations and relate them to reduction maps. Moreover, we equip a set of f-representations with a group operation which can be computed purely with baby steps, giant steps and relative distances.
We give the definition of one-dimensional infrastructures and construct baby and giant steps. Moreover, we show that one-dimensional infrastructures generalize finite cyclic groups. Finally, we give some remarks on our choice of the giant step definition.
We discuss the discrete logarithm problem, its use in cryptography, and two possible directions of generalization to other algebraic structures.
We show some kind of universal property for the Jordan decomposition of an endomorphism of a finite dimensional vector space.
We want to give a proof of the Cayley-Hamilton Theorem for all commutative rings with unity, which first reduces to the case of the field of complex numbers and then applies a topological argument.
We want to give a proof of the Fundamental Theorem of Algebra using methods from Complex Analysis, in particular Liouville's Theorem.