Some applications of decomposable form equations to resultant equations

K. Győry

Colloquium Mathematicae (1993)

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

Abstract

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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

How to cite

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Győry, 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

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  21. [21] W. M. Schmidt, Inequalities for resultants and for decomposable forms, in: Diophantine Approximation and its Applications, Academic Press, New York 1973, 235-253. 
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  25. [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

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