Related papers: Enumerating number fields
We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results…
The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting…
It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain…
Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of…
We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.
We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…
We give some bounds on edge Folkman numbers.
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
We survey algorithms and bounds for constructing planar drawings of graphs in small area.
We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.
The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.
We describe an online database of number fields which accompanies this paper The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to…
It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
The ring of integers and the discriminant are determined for number fields which are simple radical extensions.
The aim of this paper is to survey and extend results concerning bounds of the Euclidean minima of abelian number fields. In particular, we give upper bounds for the Euclidean minima of abelian number fields of prime power conductor.
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
In this paper, we give several asymptotic formulas for the number of multiplicatively dependent vectors of algebraic numbers of fixed degree, or within a fixed number field, and bounded height.