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Zbl 0780.11016
Evertse, J.H.; Györy, K.
Lower bounds for resultants. I. (English)
[J] Compos. Math. 88, No.1, 1-23 (1993). ISSN 0010-437X; ISSN 1570-5846/e

Let $F,G\in\bbfZ[x,y]$ 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 $\vert R(F,G)\vert$ 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.\par The main tools of the proofs are some results of {\it J. H. Evertse} [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)] and {\it 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.\par As a consequence of the main results the authors give applications to resultant inequalities and Thue-Mahler inequalities.
[I.Gaál (Debrecen)]

MSC 2000:: *11D75 Diophantine inequalities

Keywords: binary forms; resultant; lower bounds; rings of $S$-integers; resultant inequalities; Thue-Mahler inequalities

Citations: Zbl 0547.10008; Zbl 0554.10009

Cited in: Zbl 0973.11071 Zbl 0795.11017 Zbl 0795.11018 Zbl 0790.11026

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