## On $$S$$-integral solutions of the equation $$y^ m=f(x)$$.(English)Zbl 0552.10009

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)].
Reviewer: K. Györy

### MSC:

 11D41 Higher degree equations; Fermat’s equation

### Citations:

Zbl 0544.10013; Zbl 0544.10012; Zbl 0497.10010; Zbl 0512.10010
Full Text:

### References:

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