# Upper bounds for the degrees of decomposable forms of given discriminant

Acta Arithmetica (1994)

- Volume: 66, Issue: 3, page 261-268
- ISSN: 0065-1036

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topK. Győry. "Upper bounds for the degrees of decomposable forms of given discriminant." Acta Arithmetica 66.3 (1994): 261-268. <http://eudml.org/doc/206605>.

@article{K1994,

abstract = {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.},

author = {K. Győry},

journal = {Acta Arithmetica},

keywords = {upper bounds for degrees; decomposable form},

language = {eng},

number = {3},

pages = {261-268},

title = {Upper bounds for the degrees of decomposable forms of given discriminant},

url = {http://eudml.org/doc/206605},

volume = {66},

year = {1994},

}

TY - JOUR

AU - K. Győry

TI - Upper bounds for the degrees of decomposable forms of given discriminant

JO - Acta Arithmetica

PY - 1994

VL - 66

IS - 3

SP - 261

EP - 268

AB - 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.

LA - eng

KW - upper bounds for degrees; decomposable form

UR - http://eudml.org/doc/206605

ER -

## References

top- [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. Zbl0746.11019
- [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. Zbl0767.11017
- [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. Zbl0303.12001
- [6] I. Heller, On linear systems with integral valued solutions, Pacific J. Math. 7 (1957), 1351-1364. Zbl0079.01903

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