# Calculating “Small” Solutions of Relative Thue Equations

@article{Gal2015CalculatingS, title={Calculating “Small” Solutions of Relative Thue Equations}, author={Istv{\'a}n Ga{\'a}l}, journal={Experimental Mathematics}, year={2015}, volume={24}, pages={142 - 149} }

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… Expand

#### Topics from this paper

#### 4 Citations

Calculating “small” solutions of inhomogeneous relative Thue inequalities

- Mathematics
- Functiones et Approximatio Commentarii Mathematici
- 2021

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… Expand

Calculating Power Integral bases by Solving Relative Thue Equations

- Mathematics
- 2014

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… Expand

Power integral bases in cubic and quartic extensions of real quadratic fields

- Mathematics
- Acta Scientiarum Mathematicarum
- 2019

Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case we… Expand

Relative Thue Equations

- Physics
- Diophantine Equations and Power Integral Bases
- 2019

Let M ⊂ K be algebraic number fields, and let K = M(α) with an algebraic integer α. Let \(0\neq \mu \in {\mathbb Z}_M\). Consider the relative Thue equation
$$\displaystyle N_{K/M}(X-\alpha Y)=\mu… Expand

#### References

SHOWING 1-10 OF 22 REFERENCES

On the resolution of relative Thue equations

- Computer Science, Mathematics
- Math. Comput.
- 2002

An efficient algorithm is given for the resolution of relative Thue equations by the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst. Expand

Transcendental Number Theory

- Mathematics
- 1975

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having… Expand

Computing Power Integral Bases in Quartic Relative Extensions

- Mathematics
- 2000

Abstract 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… Expand

Factoring polynomials with rational coefficients

- Mathematics, Computer Science
- 1982

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into… Expand

On the resolution of index form equations

- Mathematics, Computer Science
- ISSAC '91
- 1991

It turned out that although for cubic fields the resolution of (I) required only about 1 minute of CPU time, already in the most simple case of quartic number fields the CPU time was too long and the solution was always found. Expand

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

- Computer Science, Mathematics
- J. Symb. Comput.
- 2000

An efficient algorithm for solving index form equations in number fields of degree 9 which are composites of cubic fields with coprime discriminants which 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. Expand

On the Resolution of Thue Inequalities

- Mathematics, Computer Science
- J. Symb. Comput.
- 1987

Let F(x, y) @? Z[x, y] be a homogenous polynomial of degree at least 3, and m @? Z. We describe a method for the resolution in (x, y) @? Z^2, |y| =< y"0 of the inequality |F(x, y)| =< m, using the… Expand

Power integral bases in parametric families of biquadratic fields

- Mathematics
- 2011

We consider two families of totally complex biquadratic fields depending on two parameters. These families were recently considered by J.G.Huard, B.K.Spearman and K.S.Williams [8]. Using our general… Expand

Improved methods for calculating vectors of short length in a lattice

- Mathematics
- 1985

The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of Z'.. in a suitable box. However, it suffices to consider… Expand

On the Resolution of Index form Equations in Sextic Fields with an Imaginary Quadratic Subfield

- Computer Science, Mathematics
- J. Symb. Comput.
- 1996

Abstract We give an efficient algorithm for the resolution of index form equations, especially for determining power integral bases, in sextic fields with an imaginary quadratic subfield. The method… Expand