Related papers: Efficient computations in central simple algebras …
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…
We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
In this paper, we give quantum algorithms for two fundamental computation problems: solving polynomial systems over finite fields and optimization where the arguments of the objective function and constraints take values from a finite field…
Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this…
Quasi-separable matrices are a class of rank-structured matriceswidely used in numerical linear algebra and of growing interestin computer algebra, with applications in e.g. the linearization ofpolynomial matrices. Various representation…
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…
The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete logarithm problem, graph isomorphism, and the shortest vector problem. The celebrated polynomial-time quantum…
The diagram algebra introduced by Brauer that describes the centralizer algebra on tensor products of the natural representation of an orthogonal group has a presentation by generators and relations that only depends on the graph of type An…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
Disentanglement of constituent factors of a sensory signal is central to perception and cognition and hence is a critical task for future artificial intelligence systems. In this paper, we present a compute engine capable of efficiently…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
A classical theorem due to Brauer and Witt implies that every simple component of the rational group algebra QG of a finite group G is Brauer equivalent to a cyclotomic algebra containing Q in its centre. The precise description of this…
We introduce Brauer algebras associated to complex reflection groups of type $G(m,p,n)$, and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…