Related papers: Note on the Ideal Exponentiation in Imaginary Quad…
Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher…
Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant…
For a cubic algebraic extension $K$ of $\mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for…
We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear…
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…
We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first…
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…
This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real…
We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even…
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…
Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher…
We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation…
Quantum computing has attracted significant interest in the optimization community because it potentially can solve classes of optimization problems faster than conventional supercomputers. Several researchers proposed quantum computing…
The growing demand for solving large-scale, data-intensive linear and conic optimization problems, particularly in applications such as artificial intelligence and machine learning, has highlighted the limitations of classical interior…
We present an alternative method for computing primary decomposition of zero-dimensional ideals over finite fields. Based upon the further decomposition of the invariant subspace of the Frobenius map acting on the quotient algebra in the…
In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are…
The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows to acknowledge errors in the implementations of quantum algorithms; on the other, it allows to…
We propose new quantum algorithms to solve the regulator and the principal ideal problem in a real-quadratic number field. We improve the algorithms proposed by Hallgren by using two different techniques. The first improvement is the usage…