The number of solutions of decomposable form equations
type: Item
instance of: scholarly article
title: The number of solutions of decomposable form equations
zbMATH Open document ID: 0851.11019
DOI: 10.1007/BF01231456
author: Jan-Hendrik Evertse
published in: Inventiones Mathematicae
publication date: 1996-11-25T00:00:00Z
full work available at URL: https://eudml.org/doc/144332
review text: Decomposable form equations, being the common generalization of several types of classical diophantine equations (among others Thue equations, norm form equations, discriminant form and index form equations) have an extensive literature. Using his famous subspace theorem, in 1972 \textit{W. M. Schmidt} [Ann. Math., II. Ser. 96, 526-551 (1972; Zbl 0245.10008)] proved the finiteness of the number of solutions of nondegenerate norm form equations. In 1977 \textit{H. P. Schlickewei} [J. Number Theory 9, 370-380 (1977; Zbl 0365.10016)], extended Schmidt's result to \(p\)-adic norm form equations, and in 1984 \textit{M. Laurent} [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)] to norm form equations over number fields. The first finiteness results on decomposable form equations were obtained by the author and \textit{K. Györy} in 1988 [Acta Arith. 50, 357-379 (1988; Zbl 0595.10013)]. In 1989 the author, the reviewer and \textit{K. Györy} [Arch. Math. 52, 337-353 (1989; Zbl 0671.10013)] showed the existence of a uniform bound \(C\) (depending only on the number of variables, the \(S\)-unit group involved, and the splitting field) such that any nondegenerate decomposable form equation (with the above parameters) can have at most \(C\) cosets of solutions. These results applied to Schmidt's subspace theorem and its \(p\)-adic generalization given by Schlickewei. In 1989 \textit{W. M. Schmidt} [Compos. Math. 69, 121-173 (1989; Zbl 0683.10027)] gave a quantitative version of his subspace theorem which enabled him to derive an explicit upper bound for the number of solutions of norm form equations with a non-degenerate module. \textit{H. P. Schlickewei} [Compos. Math. 82, 245-273 (1992; Zbl 0751.11033)] extended the quantitative subspace theorem to the number field case and obtained also an explicit upper bound for the number of solutions of \(S\)-unit equations over number fields. In 1993 \textit{K. Györy} [Publ. Math. 42, 65-101 (1993; Zbl 0792.11004)] applied this result t…
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 871780
zbMATH Keywords: \(S\)-unit equations
zbMATH Keywords: decomposable form equation
zbMATH Keywords: non-degenerate solutions
zbMATH Keywords: number of solutions
zbMATH Keywords: norm form
reviewed by: István Gaál
MaRDI profile type: Publication
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OpenAlex ID: W1999770486
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description: scientific article; zbMATH DE number 871780