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1993 | 65 | 2 | 267-275
Tytuł artykułu

Some applications of decomposable form equations to resultant equations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications of Theorems 1 and 2 are also presented to polynomials in R[X] assuming unit values at many given points in R (cf. Corollary 1) and to arithmetic progressions of given order, consisting of units of R (cf. Corollary 2). Further applications to irreducible polynomials will be given in a separate paper. Our Theorem 3 seems to be interesting in itself as well. It is deduced from some general results of Evertse and the author [3] on decomposable form equations. Since the proofs in [3] depend among other things on the Thue-Siegel-Roth-Schmidt method and its p-adic generalization
Słowa kluczowe
Rocznik
Tom
65
Numer
2
Strony
267-275
Opis fizyczny
Daty
wydano
1993
otrzymano
1993-02-01
Twórcy
autor
  • Institute of Mathematics, Lajos Kossuth University, H-4010 Debrecen, Hungary
Bibliografia
  • [1] J. H. Evertse, I. Gaál and K. Győry, On the numbers of solutions of decomposable polynomial equations, Arch. Math. (Basel) 52 (1989), 337-353.
  • [2] J. H. Evertse and K. Győry, On unit equations and decomposable form equations, J.Reine Angew. Math. 358 (1985), 6-19.
  • [3] J. H. Evertse and K. Győry, Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), 357-379.
  • [4] J. H. Evertse and K. Győry, On the numbers of solutions of weighted unit equations, Compositio Math. 66 (1988), 329-354.
  • [5] J. H. Evertse and K. Győry, Lower bounds for resultants I, to appear.
  • [6] M. Fujiwara, Some applications of a theorem of W. M. Schmidt, Michigan Math. J. 19 (1972), 315-319.
  • [7] K. Győry, On arithmetic graphs associated with integral domains, in: A Tribute to Paul Erdős, Cambridge Univ. Press, 1990, 207-222.
  • [8] K. Győry, On the numbers of families of solutions of systems of decomposable form equations, Publ. Math. Debrecen 42 (1993), 65-101.
  • [9] K. Győry, On the number of pairs of polynomials with given resultant or given semiresultant, Acta Sci. Math., to appear.
  • [10] K. Győry, Some new results connected with resultants of polynomials and binary forms, Grazer Math. Berichte 318 (1993), 17-27.
  • [11] S. Lang, Fundamentals of Diophantine Geometry, Springer, 1983.
  • [12] M. Newman, Units in arithmetic progression in an algebraic number field, Proc. Amer. Math. Soc. 43 (1974), 266-268.
  • [13] M. Newman, Consecutive units, ibid. 108 (1990), 303-306.
  • [14] A. Pethő, Application of Gröbner bases to the resolution of systems of norm equations, in: Proc. ISSAC' 91, ACM Press, 1991, 144-150.
  • [15] A. Pethő, Über kubische Ausnahmeeinheiten, Arch. Math. (Basel) 60 (1993), 146-153.
  • [16] A. Pethő, Systems of norm equations over cubic number fields, Grazer Math. Berichte 318 (1993), 111-120.
  • [17] H. P. Schlickewei, Inequalities for decomposable forms, Astérisque 41-42 (1977), 267-271.
  • [18] H. P. Schlickewei, The p-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel) 29 (1977), 267-270.
  • [19] H. P. Schlickewei, S-unit equations over number fields, Invent. Math. 102 (1990), 95-107.
  • [20] H. P. Schlickewei, The quantitative Subspace Theorem for number fields, Compositio Math. 82 (1992), 245-273.
  • [21] W. M. Schmidt, Inequalities for resultants and for decomposable forms, in: Diophantine Approximation and its Applications, Academic Press, New York 1973, 235-253.
  • [22] W. M. Schmidt, Simultaneous approximation to algebraic numbers by elements of a number field, Monatsh. Math. 79 (1975), 55-66.
  • [23] W. M. Schmidt, The subspace theorem in diophantine approximations, Compositio Math. 69 (1989), 121-173.
  • [24] H. Weber, Lehrbuch der Algebra, Erster Band, Verlag Vieweg, Braunschweig 1898.
  • [25] E. Wirsing, On approximations of algebraic numbers by algebraic numbers of bounded degree, in: Proc. Sympos. Pure Math. 20, Amer. Math. Soc., Providence 1971, 213-247.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv65i2p267bwm
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