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Effective results for Diophantine equations over finitely generated domains

100%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
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2014
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tom 163
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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.
2
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Finiteness results for Diophantine triples with repdigit values

100%
Bérczes A., Luca F., Pink I., Ziegler V.
Acta Arithmetica
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2016
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tom 172
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nr 2
133-148
EN
Let g ≥ 2 be an integer and $𝓡_g ⊂ ℕ$ be the set of repdigits in base g. Let $𝓓_g$ be the set of Diophantine triples with values in $𝓡_g$; that is, $𝓓_g$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set $𝓡_g$. We prove effective finiteness results for the set $𝓓_g$.
3
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On the number of pairs of binary forms with given degree and given resultant

100%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
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2007
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tom 128
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nr 1
19-54
4
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Effective results for linear equations in two unknowns from a multiplicative division group

100%
Bérczes A., Evertse J.-H., Győry K.
Acta Arithmetica
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2009
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tom 136
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nr 4
331-349
5
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On the number of equivalence classes of binary forms of given degree and given discriminant

100%
Bérczes A., Evertse J.-H., Győry K. a.
Acta Arithmetica
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2004
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tom 113
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nr 4
363-399
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