Related papers: Computing in arithmetic groups with Voronoi's algo…
The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are…
We provide an algorithm for computing semi-Fourier sequences for expressions constructed from arithmetic operations, exponentiations and integrations. The semi-Fourier sequence is a relaxed version of Fourier sequence for polynomials…
Given a basic compact semi-algebraic set $\K\subset\R^n$, we introduce a methodology that generates a sequence converging to the volume of $\K$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear…
Some aspects of Computer Algebra (notably Computation Group Theory and Computational Number Theory) have some good databases of examples, typically of the form "all the X up to size n". But most of the others, especially on the polynomial…
The method is introduced for fast data processing by reducing the probability amplitudes of undesirable elements. The algorithm has a mathematical description and circuit implementation on a quantum processor. The idea is to make a quick…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
In this talk we introduce several topics in combinatorial number theory which are related to groups; the topics include combinatorial aspects of covers of groups by cosets, and also restricted sumsets and zero-sum problems on abelian…
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.
In this paper, we present an improved methodology to compute $\omega$-invariant of numerical semigroup. The approach is based on adapting a recent resolution method for optimizing a linear function over the set of efficient solutions of a…
The present paper shows meta-programming turn programming, which is rich enough to express arbitrary arithmetic computations. We demonstrate a type system that implements Peano arithmetics, slightly generalized to negative numbers. Certain…
Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…
Let z be a primitive fifth root of unity and let F be the cyclotomic field F=Q(z). Let O be the ring of integers. We compute the Voronoi polyhedron of binary Hermitian forms over F and classify GL_2(O)-conjugacy classes of perfect forms.…
The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
We introduce generalised orbit algebras. The purpose here is to measure how some combinatorial properties can characterize the action of a group of permutations on the subsets. The similarity with orbit algebras is such that it took the…
We are interested in formulas for the number of elements in certain classes of numerical semigroups
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box, but the aim is to compute function value for arbitrary input using as few queries as possible. In this paper we…
Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases,…