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Index form equations in quintic fields

István GaálKálmán Győry — 1999

Acta Arithmetica

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...

Effective results for Diophantine equations over finitely generated domains

Attila BérczesJan-Hendrik EvertseKálmán Győry — 2014

Acta Arithmetica

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 ) = δ y m in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and m 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...

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