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Grover's algorithm is one of the pioneering demonstrations of the advantages of quantum computing over its classical counterpart, providing - at most - a quadratic speed-up over the classical solution for unstructured database search. The…

The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight…

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…

In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation…

Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the…

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…

A central task in the field of quantum computing is to find applications where quantum computer could provide exponential speedup over any classical computer. Machine learning represents an important field with broad applications where…

Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by…

We present here an overview of the hypermatrix spectral decomposition deduced from the Bhattacharya-Mesner hypermatrix algebra. We describe necessary and sufficient conditions for the existence of a spectral decomposition. We further extend…

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor…

Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection…

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper we consider the related problems of computing all minimal overorders…

The reduction algorithm is used to compute reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among…

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real…

A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more…

The control of quantum phenomena is a topic that has carried out many challenging problems. Among others, the Hamiltonian identification, i.e, the inverse problem associated with the unknown features of a quantum system is still an open…

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 work is to reduce the complexity of the available algorithms for computing the generator sets of a semigroup ideal by using the Hermite normal form. In order to achieve it we introduce the concept of decomposable semigroup.…

The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…