Relative Thue Equations Research Articles

Calculating “small” solutions of inhomogeneous relative Thue inequalities

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the “small” solutions of these equations. Under “small” solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain “totally real” relative Thue equations can be reduced to absolute Thue equations (equations over $\mathbb{Z}$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating “small” solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the “totally real” case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

CALCULATING RELATIVE POWER INTEGRAL BASES IN TOTALLY COMPLEX QUARTIC EXTENSIONS OF TOTALLY REAL FIELDS

Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions of complex quartic fields. In the present paper we consider power integral bases in relative extensions of totally real fields. Totally complex quartic extensions of totally real number fields seems to be the most simple case, that we detail here. As we shall see, even in this case we have to overcome several unexpected difficulties, which we can, however solve by properly (but not trivially) adjusting standard methods. We demonstrate our general algorithm on an explicit example. We describe how the general methods for solving relative index form equations in quartic relative extensions are modified in this case. As a byproduct we show that relative Thue equations in totally complex extensions of totally real fields can only have small solutions, and we construct a special method for the enumeration of small solutions of special unit equations. These statements can be applied to other diophantine problems, as well.

Calculating “Small” Solutions of Relative Thue Equations

Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions. On the other hand, to compute all solutions (i.e., to prove the nonexistence of large solutions) is a time-consuming procedure. Therefore, it is useful to have a fast algorithm to calculate the “small” solutions, especially if “small” means less than, e.g., 10100. Such an algorithm was constructed by A. Pethö in 1987 based on continued fractions.In the present paper, we construct a similar type of fast algorithm to calculate “small” solutions of relative Thue equations. Our method is based on the LLL reduction algorithm. We illustrate the method with explicit examples. The algorithm has several applications.

Calculating Power Integral bases by Solving Relative Thue Equations

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

On a parameterized family of relative Thue equations

On a parameterized family of relative Thue equations

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

On the resolution of relative Thue equations

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.The method is illustrated by numerical examples.

Power Integral Bases in Cubic Relative Extensions

We give an efficient algorithm for computing relative power integral bases in cubic relative extensions. The problem leads to solving relative Thue equations as described by [Gaál and Pohst 1999] using the enumeration method of [Wildanger 1997]. The article is illustrated by examples of relative cubic extensions of quintic and sextic fields which emphasizes the power of the method. This is the first case that unit equations of 12 unknown exponents are completely solved. The experiences of our computations may be useful for other related calculations, as

Computing Power Integral Bases in Quartic Relative Extensions

We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaál, A. Pethő, and M. Pohst, 1993, J. Symbolic Comput.16, 563–584; 1996, J. Number Theory57, 90–104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaál and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples.

Solving Index Form Equations in Fields of Degree 9 with Cubic Subfields

We describe an efficient algorithm for solving index form equations in number fields of degree 9 which are composites of cubic fields with coprime discriminants. We develop the algorithm in detail for the case of complex cubic fields, but the main steps of the procedure are also applicable for other cases. Our most important tool is the main theorem of a recent paper of Gaál (1998a). In view of this result the index form equation in the ninth degree field implies relative index form equations over the subfields. In our case these equations are cubic relative Thue equations over cubic fields. The main purpose of the paper is to show that this approach is much more efficient than the direct method, which consists of reducing the index form equation to unit equations over the normal closure of the original field. At the end of the paper we describe our computational experience. Many ideas of the paper can be applied to develop fast algorithms for solving index form equations in other types of higher degree fields which are composites of subfields.

Computing all power integral bases in orders of totally real cyclic sextic number fields

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.