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Lower bounds for resultants. I. (English) Zbl 0780.11016
Let \(F,G\in\mathbb{Z}[x,y]\) be binary forms of degrees at least three, and consider the resultant \(R(F,G)\) of these forms. Denote by \(L\) the splitting field of \(F\cdot G\) and assume, that \(F\cdot G\) is square free. The main results of the paper give lower bounds for \(| R(F,G)|\) depending on \(L\) and the degrees and the discriminants of \(F\), \(G\). These main theorems are formulated and proved in a more general form, over the rings of \(S\)-integers of algebraic number fields.
The main tools of the proofs are some results of J. H. Evertse [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)] and M. Laurent [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)] on \(S\)-unit equations in several unknowns. These results of Evertse and Laurent are based on H. P. Schlickewei’s \(p\)-adic generalization of W. M. Schmidt’s subspace theorem, and are therefore ineffective. This is the reason, why the results of the present paper are semi-effective, that means, the inequalities include also ineffective constants.
As a consequence of the main results the authors give applications to resultant inequalities and Thue-Mahler inequalities.
Reviewer: I.Gaál (Debrecen)

MSC:
11D75 Diophantine inequalities
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References:
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