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1994 | 66 | 3 | 261-268
Tytuł artykułu

Upper bounds for the degrees of decomposable forms of given discriminant

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Języki publikacji
EN
Abstrakty
EN
1. Introduction. In our paper [5] a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author [3] to decomposable forms in n ≥ 2 variables. The bound obtained in [3] depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish an improvement of the bound of [3] which is already best possible for every n ≥ 2. Moreover, all the squarefree decomposable forms in n variables over ℤ will be determined for which our bound cannot be further sharpened. In the proof we shall use some results and arguments of [5] and [3] and two theorems of Heller [6] on linear systems with integral valued solutions.
Słowa kluczowe
Czasopismo
Rocznik
Tom
66
Numer
3
Strony
261-268
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-08-30
Twórcy
autor
  • Institute of Mathematics, Lajos Kossuth University, H-4010 Debrecen, Hungary
Bibliografia
  • [1] Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York, 1967.
  • [2] J. H. Evertse and K. Győry, Effective finiteness theorems for decomposable forms of given discriminant, Acta Arith. 60 (1992), 233-277.
  • [3] J. H. Evertse and K. Győry, Discriminants of decomposable forms, in: Analytic and Probabilistic Methods in Number Theory, F. Schweiger and E. Manstavičius (eds.), VSP Int. Science Publ., Zeist, 1992, 39-56.
  • [4] I. M. Gelfand, A. V. Zelevinsky and M. M. Karpanov, On discriminants of polynomials of several variables, Funktsional. Anal. i Prilozhen. 24 (1990), 1-4 (in Russian).
  • [5] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné II, Publ. Math. Debrecen 21 (1974), 125-144.
  • [6] I. Heller, On linear systems with integral valued solutions, Pacific J. Math. 7 (1957), 1351-1364.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-aav66i3p261bwm
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