Related papers: The Euclidean Algorithm in Cubic Number Fields
Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over an…
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training…
We study elements of small norm in quadratic fields of Riachaud-Degert type.
We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.
In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2,3) with absolute discriminant less than a certain bound. The…
Primal and dual algorithms are developed for solving the $n$-dimensional convex optimization problem of finding the Euclidean ball of minimum radius that covers $m$ given Euclidean balls, each with a given center and radius. Each algorithm…
In this paper we intend to show that certain integers do not occur as the norms of principal ideals in a family of cubic fields studied by Cohn, Shanks, and Ennola. These results will simplify the construction of certain unramified…
We obtain lower bound of caliber number of real quadratic field $K=\FQ(\sqrt{d})$ using splitting primes in $K$. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if $d$ is not 5…
We describe an algorithm for listing all elements of bounded height in a given number field.
Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection,…
We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…
Let $K$ be a cyclic cubic field and $\mathcal{O}_K$ be its ring of integers. In this note we prove that all cyclic cubic number fields with conductors in the interval $ [73, 11971]$ and with class number one are Euclidean.
We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.
Given a real quadratic integer $u=A+B\sqrt{D}$ with cubic norm, we identify all the classes in a related form class group that represent primes $p$ for which $u$ is a cubic residue mod $p$. A special case of this result was conjectured in a…
Let $K$ and $K'$ be arithmetically equivalent number fields, both of degree $d$. We prove that $K$ and $K'$ have the same successive minima, up to a constant depending only on $d$. We give examples showing that one cannot expect equality.
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real…
We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…
This paper deals with the quadratic integers of small norms and asserts that in some sense R >> (log D)^2 is true for almost all real quadratic number fields. (A few errata is corrected.)
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…