Related papers: Computational details on the disproof of modularit…
Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among…
This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…
We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log(A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal…
In this note we give a wellfoundedness proof of a computable notation system for first-order reflection.
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with…
During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^\lambda for the symmetric group, using the computer algebra program Magma. The complexity…
This work studies the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.
We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple…
Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a…
A method for computing the multigraded Hilbert depth of a module was presented in [16]. In this paper we improve the method and we introduce an effective algorithm for performing the computations. In a particular case, the algorithm may…
This is a slightly revised version of the author's 2010 diploma thesis. It is concerned with the interplay between real multiplication on Jacobian varieties, as the title suggests, and complex geodesics in the moduli space of curves. Large…
The computation of the normaliser of a permutation group in the full symmetric group is an important and hard problem in computational group theory. This article reports on an algorithm that builds a descending chain of overgroups to…
Differential graded (DG) commutative algebra provides powerful techniques for proving theorems about modules over commutative rings. These notes are a somewhat colloquial introduction to these techniques. In order to provide some motivation…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
This is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of…
The aim of this paper is to introduce and study graded and filtered gamma rings and gamma modules. We prove that the filtered $\Gamma$-ring (module) is a generalization of the notion of graded ring (module). Also, we construct a graded…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…