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Effective finiteness results for binary forms with given discriminant. (English) Zbl 0746.11020

B. J. Birch and J. R. Merriman [Proc. Lond. Math. Soc., III. Ser. 24, 385-394 (1972; Zbl 0248.12002)] proved that for arbitrary \(r\geq 4\), the are only finitely many \(\mathbb{Z}\)-equivalence classes of binary forms in \(\mathbb{Z}[X,Y]\) of degree \(r\) and given discriminant. Here equivalence is defined by transformations in \(GL(2,\mathbb{Z})\). They extended their result to binary forms whose coefficients belong to the ring of \(S\)-integers of an algebraic number field. Their proof was ineffective. The main tool in the proof was the finiteness of the number of solutions of the so-called unit equation \(\alpha x+\beta y=1\) in units \(x,y\) of the ring of integers of some given algebraic number field.
The authors give an effective proof of the results of Birch and Merriman on binary forms with \(S\)-integral coefficients. Further, they give applications of these results to binary forms, algebraic numbers and discriminant form equations. Their results are formulated in a quantitative form.
Reviewer: B.Richter (Berlin)

MSC:

11D57 Multiplicative and norm form equations
11E76 Forms of degree higher than two
11D75 Diophantine inequalities

Citations:

Zbl 0248.12002
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References:

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