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Some remarks on the \(S\)-unit equation in function fields. (English) Zbl 0786.11019

Let \(R\) denote a localization of \(\mathbb{Z}\) with respect to finitely many primes and \(A_ R\) the ring of algebraic integers over \(R\). For a number field \(k\) denote \(N_ k(\alpha)\) the norm over \(\mathbb{Q}\) of the element \(\alpha\) of \(k\). The main result of this paper is the following theorem.
Let \(\alpha_ 1,\dots, \alpha_ r\) be pairwise non-conjugate algebraic numbers. Consider the set \(K\) of all number fields \(k\) such that \[ \sum_{v=1}^ r [k(\alpha_ v):k]\geq 2n[k:\mathbb{Q}]+1 \] holds with a positive integer \(n\). Then there exist modulo \(A_ R^*\) only finitely many \((x_ 0,\dots,x_ n)\in A_ R^{n+1}\) with \((x_ 0,\dots, x_ n)\in k^{n+1}\) for a \(k\in K\) and \[ N_{k(\alpha_ v)} \bigl( \sum_{u=0}^ n x_ u \alpha_ v^ u\bigr) \in R^* \qquad \text{for} \qquad 1\leq v\leq r. \] An interesting consequence of the main result is that if \(\alpha\) is an algebraic integer of degree larger than \(2ng\) then \(\bigcup_{[k:\mathbb{Q}]\leq \sqrt{g}} (\alpha^ 0,\dots, \alpha^ n)k\) does not contain a subfield \(K\) with \([K:\mathbb{Q}]>g\).

MSC:

11D57 Multiplicative and norm form equations
11R58 Arithmetic theory of algebraic function fields
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