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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
Full Text:
##### References:
 [1] B.J. Birch and J.R. Merriman , Finiteness theorems for binary forms with given discriminant , Proc. London Math. Soc. 25 (1972) 385-394. · Zbl 0248.12002 · doi:10.1112/plms/s3-24.3.385 [2] J.H. Evertse , On equations in S-units and the Thue-Mahler equation , Invent. Math. 75 (1984), 561-584. · Zbl 0521.10015 · doi:10.1007/BF01388644 · eudml:143111 [3] J.H. Evertse , On sums of S-units and linear recurrences , Compositio Math. 53 (1984) 225-244. · Zbl 0547.10008 · numdam:CM_1984__53_2_225_0 · eudml:89685 [4] J.H. Evertse and K. Györy , Thue-Mahler equations with a small number of solutions , J. Reine Angew. Math. 399 (1989) 60-80. · Zbl 0675.10009 · crelle:GDZPPN002206811 · eudml:153154 [5] J.H. Evertse and K. Györy , Effective finiteness results for binary forms with given discriminant , Compositio Math. 79 (1991) 169-204. · Zbl 0746.11020 · numdam:CM_1991__79_2_169_0 · eudml:90103 [6] J H. Evertse, K. Györy, C. L. Stewart and R. Tijdeman , On S-unit equations in two unknowns , Invent. Math. 92 (1988), 461-477. · Zbl 0662.10012 · doi:10.1007/BF01393743 · eudml:143578 [7] K. Györy , Sur les polynômes à coefficients entiers et de discriminant donné , Acta Arith. 23 (1973) 419-426. · Zbl 0269.12001 · eudml:205206 [8] K. Györy , On polynomials with integer coefficients and given discriminant, V, p-adic generalizations , Acta Math. Acad. Sci. Hungar. 32 (1978), 175-190. · Zbl 0402.10053 · doi:10.1007/BF01902212 [9] K. Györy , On arithmetic graphs associated with integral domains , in: A Tribute to Paul Erdös (eds. A. Baker, B. Bollobás, A. Hajnal), pp. 207-222. Cambridge University Press, 1990. · Zbl 0727.11039 [10] K. Györy , On the number of pairs of polynomials with given resultant or given semi-resultant , to appear. · Zbl 0798.11043 [11] M. Laurent , Equations diophantiennes exponentielles , Invent. Math. 78 (1984) 299-327. · Zbl 0554.10009 · doi:10.1007/BF01388597 · eudml:143175 [12] H.P. Schlickewei , The p-adic Thue-Siegel-Roth-Schmidt theorem , Archiv der Math. 29 (1977) 267-270. · Zbl 0365.10026 · doi:10.1007/BF01220404 [13] W.M. Schmidt , Inequalities for resultants and for decomposable forms , in: Diophantine Approximation and its Applications (ed. C. F. Osgood), pp. 235-253, Academic Press, New York, 1973. · Zbl 0267.10023 [14] W.M. Schmidt , Diophantine Approximation , Lecture Notes in Math. 785, Springer-Verlag, 1980. · Zbl 0421.10019 [15] E. Wirsing , On approximations of algebraic numbers by algebraic numbers of bounded degree , in: Proc. Symp. Pure Math. 20 (1969 Number Theory Institute ; ed. D. J. Lewis), pp. 213-247, Amer. Math. Soc., Providence, 1971. · Zbl 0223.10017
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