# Some applications of decomposable form equations to resultant equations

Colloquium Mathematicae (1993)

- Volume: 65, Issue: 2, page 267-275
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topGyőry, K.. "Some applications of decomposable form equations to resultant equations." Colloquium Mathematicae 65.2 (1993): 267-275. <http://eudml.org/doc/210220>.

@article{Győry1993,

abstract = {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},

author = {Győry, K.},

journal = {Colloquium Mathematicae},

keywords = {resultants; decomposable form equation},

language = {eng},

number = {2},

pages = {267-275},

title = {Some applications of decomposable form equations to resultant equations},

url = {http://eudml.org/doc/210220},

volume = {65},

year = {1993},

}

TY - JOUR

AU - Győry, K.

TI - Some applications of decomposable form equations to resultant equations

JO - Colloquium Mathematicae

PY - 1993

VL - 65

IS - 2

SP - 267

EP - 275

AB - 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

LA - eng

KW - resultants; decomposable form equation

UR - http://eudml.org/doc/210220

ER -

## References

top- [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. Zbl0655.10017
- [2] J. H. Evertse and K. Győry, On unit equations and decomposable form equations, J.Reine Angew. Math. 358 (1985), 6-19. Zbl0552.10010
- [3] J. H. Evertse and K. Győry, Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), 357-379. Zbl0595.10013
- [4] J. H. Evertse and K. Győry, On the numbers of solutions of weighted unit equations, Compositio Math. 66 (1988), 329-354. Zbl0644.10015
- [5] J. H. Evertse and K. Győry, Lower bounds for resultants I, to appear. Zbl0780.11016
- [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. Zbl0792.11004
- [9] K. Győry, On the number of pairs of polynomials with given resultant or given semiresultant, Acta Sci. Math., to appear. Zbl0798.11043
- [10] K. Győry, Some new results connected with resultants of polynomials and binary forms, Grazer Math. Berichte 318 (1993), 17-27. Zbl0795.11018
- [11] S. Lang, Fundamentals of Diophantine Geometry, Springer, 1983. Zbl0528.14013
- [12] M. Newman, Units in arithmetic progression in an algebraic number field, Proc. Amer. Math. Soc. 43 (1974), 266-268. Zbl0285.12011
- [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. Zbl0922.11116
- [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. Zbl0804.11027
- [17] H. P. Schlickewei, Inequalities for decomposable forms, Astérisque 41-42 (1977), 267-271. Zbl0365.10018
- [18] H. P. Schlickewei, The p-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel) 29 (1977), 267-270. Zbl0365.10026
- [19] H. P. Schlickewei, S-unit equations over number fields, Invent. Math. 102 (1990), 95-107. Zbl0711.11017
- [20] H. P. Schlickewei, The quantitative Subspace Theorem for number fields, Compositio Math. 82 (1992), 245-273. Zbl0751.11033
- [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. Zbl0317.10042
- [23] W. M. Schmidt, The subspace theorem in diophantine approximations, Compositio Math. 69 (1989), 121-173. Zbl0683.10027
- [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. Zbl0223.10017

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.