On \(S\)-integral solutions of the equation \(y^ m=f(x)\)
Let \(K\) be an algebraic number field with ring of integers \({\mathcal O}_ K\), let \(f(x)\in {\mathcal O}_ K[X]\) and suppose \(f(x)=a_ 0\prod^{n}_{i=1}(x-\alpha_ i)^{r_ i}\) with \(a_ 0\neq 0\), \(\alpha_ i\neq \alpha_ j\) for \(i\neq j\). Let \(m\geq 2\) be an integer and put \(t_ i=m/(m,r_ i)\), \(i=1,\ldots,r\). Let \(L\) be a finite extension field of \(K\), let \({\mathfrak p}_ 1,\ldots,{\mathfrak p}_ s\) \((s\geq 0)\) be distinct prime ideals in \(L\), let \(S=\{{\mathfrak p}_ 1,\ldots,{\mathfrak p}_ s\}\), let \({\mathcal O}_ s\) denote the ring of \(S\)-integers in \(L\) and let \(P=\max N({\mathfrak p}_ i)\) (with \(P=1\) if \(s=0)\). The following theorem is proved: Suppose that \((t_ 1,\ldots,t_ n)\) is not a permutation of either of the \(n\)-tuples \((t,1,1,\ldots,1)\) and \((2,2,1,1,\ldots,1)\). Then all solutions \((x,y)\in {\mathcal O}^ 2_ s\) of the equation \(y^ m=f(x)\) satisfy \[ \max (H(x),H(y))<\exp \exp \{cP^ 2(s+1)^ 3\}, \] where \(c\) is an effectively computable constant depending only on \(L\), \(f(x)\) and \(m\). Applications of this theorem are given in two recent papers of the author [Acta Arith. 44, No. 2, 99--107 (1984; Zbl 0497.10010); and ibid. 44, No. 4, 357--363 (1984; Zbl 0512.10010)].
scientific article; zbMATH DE number 3879008