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1
Content available remote

Bounds for the solutions of S-unit equations and decomposable form equations

100%
Győry K., Yu K.
Acta Arithmetica
|
2006
|
tom 123
|
nr 1
9-41
2
Content available remote

Bounds for the solutions of Thue-Mahler equations and norm form equations

100%
Bugeaud Y., Győry K.
Acta Arithmetica
|
1996
|
tom 74
|
nr 3
273-292
3
Content available remote

Index form equations in quintic fields

100%
Gaál I., Győry K.
Acta Arithmetica
|
1999
|
tom 89
|
nr 4
379-396
EN
The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit equations in which the unknown units are elements of unit groups generated by much fewer generators. On the other hand, Wildanger [32] worked out an efficient enumeration algorithm that makes it feasible to solve unit equations even if the rank of the unit group is ten. Combining these developments we describe an algorithm to solve completely index form equations in quintic fields. The method is illustrated by numerical examples: we computed all power integral bases in totally real quintic fields with Galois group S₅.
4
Content available remote

Bounds for the solutions of unit equations

100%
Bugeaud Y., Győry K.
Acta Arithmetica
|
1996
|
tom 74
|
nr 1
67-80
5
Content available remote

Effective results for Diophantine equations over finitely generated domains

81%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
|
2014
|
tom 163
|
nr 1
71-100
EN
Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f(x) = δy^m$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖{0} and $m ∈ ℤ_{≥ 2}$. Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, m. These results imply that the solutions of these equations can be determined in principle. Further, we consider the Schinzel-Tijdeman equation $f(x) = δy^m$ where x,y ∈ A and $m ∈ ℤ_{≥2}$ are the unknowns and give an effective upper bound for m. Our results extend earlier work of Győry, Brindza and Végső, where the equations mentioned above were considered only for a restricted class of finitely generated domains.
6
Content available remote

On the number of pairs of binary forms with given degree and given resultant

81%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
|
2007
|
tom 128
|
nr 1
19-54
7
Content available remote

On power values of pyramidal numbers, I

81%
Dujella A., Győry K., Pintér Á.
Acta Arithmetica
|
2012
|
tom 155
|
nr 2
217-226
8
Content available remote

Effective results for linear equations in two unknowns from a multiplicative division group

81%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
|
2009
|
tom 136
|
nr 4
331-349
9
Content available remote

Index form equations in sextic fields: a hard computation

81%
Bilu Y., Gaál I., Győry K.
Acta Arithmetica
|
2004
|
tom 115
|
nr 1
85-96
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