Solving resultant form equations over number fields

Gaál, István; Pohst, Michael

doi:10.1090/s0025-5718-08-02141-8

Cited by 3 publications

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References 12 publications

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“…Given f , using the theory of unit equations I.Gaál and M.Pohst [18] gave an algorithm to determine all g having roots in a fixed number field.…”

Section: Application To Resultant Type Equationsmentioning

confidence: 99%

Calculating “Small” Solutions of Relative Thue Equations

Gaál

2015

Experimental Mathematics

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Diophantine equations can often be reduced to various types of classical Thue equations [20], [1]. These equations usually have only very small solutions, on the other hand to compute all solutions (i.e. to prove the non-existence of large solutions) is a time consuming procedure. Therefore it is very practical to have a fast algorithm to calculate the "small" solutions, especially if "small" means less than e.g. 10 100 . Such an algorithm was constructed by A.Pethő [17] 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.

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“…Given f , using the theory of unit equations I.Gaál and M.Pohst [18] gave an algorithm to determine all g having roots in a fixed number field.…”

Section: Application To Resultant Type Equationsmentioning

confidence: 99%

Calculating “Small” Solutions of Relative Thue Equations

Gaál

2015

Experimental Mathematics

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On the equation Resx(P(x),x2 + sx + t) = a

Alkabouss

Garici

Larone

2018

Int. J. Number Theory

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Calculating “small” solutions of inhomogeneous relative Thue inequalities

Gaál¹

2021

Funct. Approx. Comment. Math.

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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 ≤ 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 Z).As a common generalization of the above results, in our paper we develop a fast algorithm for calculating "small" solutions (say with sizes ≤ 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.

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Solving resultant form equations over number fields

Abstract: Abstract. We give an efficient algorithm for solving resultant form equations over number fields. This is the first time that such equations are completely solved by reducing them to unit equations in two variables.

Cited by 3 publications

References 12 publications

Calculating “Small” Solutions of Relative Thue Equations

Calculating “Small” Solutions of Relative Thue Equations

On the equation Resx(P(x),x2 + sx + t) = a

Calculating “small” solutions of inhomogeneous relative Thue inequalities

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