Related papers: Computations in Cubic Function Fields of Character…

This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely…

We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…

We use a function field version of the circle method to prove that a positive proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of minimal degree from $\mathbb{F}_q[t]$, assuming a suitable form of the…

We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an…

computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that…

We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric.

The purpose of this paper is to propose an efficient method to compute the automorphism group of an arbitrary hyperelliptic function field (genus>1) over a given ground field of characteristic >2 as well as over its algebraic extensions.

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such…

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain…

In this paper, we analyze the planar cubic Alternative curve to determine the conditions for convex, loops, cusps and inflection points. Thus cubic curve is represented by linear combination of three control points and basis function that…

We give an asymptotic formula for class numbers of orders in cubic number fields.

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was…

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for…

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 n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an…