Norm form equations. I
type: Item
instance of: scholarly article
title: Norm form equations. I
zbMATH Open document ID: 0578.10021
DOI: 10.1016/0022-314X(86)90069-7
author: R. C. Mason
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(L\) be an arbitrary finite extension of the rational function field \(k(z)\) over an algebraically closed field \(k\) of characteristic zero, let \(\mathcal O\) denote the ring of elements of \(L\) integral over \(k[z]\), and let \(M\) be a finitely generated \({\mathcal O}\)-module lying in a finite extension \(K\) of \(L\). Each valuation \(v\) on \(K\) is defined as the order of vanishing of the Laurent expansion in a local parameter \(z_ v\), and those valuations with \(v(z)<0\) are termed infinite. The height \(H(f)\) of any element \(f\) in \(K\) is defined by \(H(f)=-\sum_{v}\min (0,v(f)),\) and the height \(H({\mathcal F})\) of any finite set \({\mathcal F}\) of elements of \(K\) is defined by \(H({\mathcal F})=-\sum_{v}\min (0,v(f); f\in {\mathcal F}).\) The main results of this paper read as follows. Suppose that \(K\) be a Galois extension of \(L\). Then, for each \(c\) in \(L\) all the solutions \(x\) in \(M\) of the equation (*) \(\text{Norm}_{K/L}(x)=c\) may be determined effectively. Moreover, these solutions are finite in number. Each solution \(x\) in \(M\) of (*) satisfies \[ H(x)\leq (2d)^{3n-3} 2^{n^2} (H+H(c)+g_K+r_K+1), \] where \(H\) denotes the height of an \(\mathcal O\)-basis \(x_1,\dots, x_n\) of \(M\), and \(d\), \(g_K\), \(r_K\) denote the degree \([K:L]\), the genus of \(K/k\), and the number of infinite valuations on \(K\), respectively.
reviewed by: Saburô Uchiyama
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3924881
zbMATH Keywords: unit equation
zbMATH Keywords: norm form equations
zbMATH Keywords: rational function field
zbMATH Keywords: Galois extension
zbMATH Keywords: height
zbMATH Keywords: infinite valuations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3924881