Related papers: Algorithm for computing the factor ring of an idea…
We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the…
We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued…
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…
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.…
Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…
The principal ideal problem constitutes a fundamental problem in algebraic number theory and has attracted significant attention due to its applications in ideal lattice based cryptosystems. Efficient quantum algorithm has been found to…
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…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity…
Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension…
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…
Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], we show how to compute the Hermite form H of A, and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in terms of…
Motivated by Carmichael numbers, we say that a finite ring $R$ is a Carmichael ring if $a^{|R|}=a$ for any $a \in R$. We then call an ideal $I$ of a ring $R$ as a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of…
We study those integral domains in which every proper ideal can be written as an invertible ideal multiplied by a nonempty product of proper radical ideals.
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring…
We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness…
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
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…
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
In the present article we shown a formula to compute the number of all matrices over the finite field $F$ whit prescribed eigenvalues. Using this formula we obtain one inequality for the number of $(k+1)$-potent elements over finite rings.
We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in…