相关论文: Algorithms in algebraic number theory

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…

Finding inclusion-minimal "hitting sets" for a given collection of sets is a fundamental combinatorial problem with applications in domains as diverse as Boolean algebra, computational biology, and data mining. Much of the algorithmic…

The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on…

Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these…

We will pursue a way of building up an algebraic structure that involves, in a mathematical abstract way, the well known Grassmann variables. The problem arises when we tried to understand the grassmannian polynomial expansion on the scope…

In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the…

Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…

In a wide array of areas, algorithms are matching and surpassing the performance of human experts, leading to consideration of the roles of human judgment and algorithmic prediction in these domains. The discussion around these…

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this…

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a…

We propose an approach to image processing related to algebraic operators acting in the space of images. In view of the interest in the applications in optics and computer science, mathematical aspects of the paper have been simplified as…

Decision theories offer principled methods for making choices under various types of uncertainty. Algorithms that implement these theories have been successfully applied to a wide range of real-world problems, including materials and drug…

Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to…

Algebraic quantum field theory is an approach to relativistic quantum physics, notably the theory of elementary particles, which complements other modern developments in this field. It is particularly powerful for structural analysis but…

We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$,…

The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…

In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with…

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…

In this paper we relate a number of parsing algorithms which have been developed in very different areas of parsing theory, and which include deterministic algorithms, tabular algorithms, and a parallel algorithm. We show that these…