Norm form equations. I. (English) Zbl 0578.10021

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.


11D57 Multiplicative and norm form equations
12J25 Non-Archimedean valued fields
11R58 Arithmetic theory of algebraic function fields
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