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Discriminants of decomposable forms. (English) Zbl 0767.11017

Analytic and probabilistic methods in number theory. Proc. Int. Conf. in Honour of J. Kubilius, Palanga/Lith. 1991, New Trends Probab. Stat. 2, 39-56 (1992).
The paper extends the earlier results of K. Győry [Publ. Math. 21, 125–144 (1974; Zbl 0303.12001)] on discriminants of binary forms to the case of decomposable forms in \(n\geq 2\) variables. In the main result (Theorem 1) of the paper it is shown, that if \(F\in\mathbb{Z}[X_1,\ldots,X_n]\) is a square-free decomposable form of rank \(m\) and degree \(r\), then
\[ r\leq 2^m-1+\frac{m}{\log 3}\log D_F \]
and if \(F\) is not divisible by a linear form with coefficients in \(\mathbb{Q}\), then
\[ r\leq {m\over {\log 3}}\log D_ F \]
where \(D_ F\) denotes the discriminant of \(F\). In view of a result of the authors [Acta Arith. 60, 233–277 (1992; Zbl 0746.11019)] this theorem yields, that for every \(n\geq 2\) and \(D\geq 0\), there are only finitely many equivalence classes of primitive, square-free forms in \(\mathbb{Z}[X_1,\dots,X_n]\) of maximal rank and discriminant \(D\), and a full set of representatives of these classes can be effectively determined.
In Section 2 the authors consider decomposable forms over Dedekind domains and in Section 3 an analogue of the main result is proved for decomposable forms over \(S\)-integers in number fields.
[For the entire collection see Zbl 0754.00023.]

MSC:

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