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A “Hardy-Littlewood” approach to the $$S$$-unit equation. (English) Zbl 0686.10013
Let $$K$$ be an algebraic number field of degree $$n$$ over $$\mathbb Q$$. Let $${\mathfrak x}=(x_ 0,\ldots,x_ n)$$ be an element of the projective space $$P^ n(K)$$. Define the projective height of $${\mathfrak x}$$ to be $$H({\mathfrak x}):=\prod_{v}\max \{| x_ 0|_ v,\ldots,| x_ n|_ v\}$$, the product being taken over all the valuations $$v$$ on $$K$$, and let $$N({\mathfrak x}):=\prod_{v}| c_ 0x_ 0+\ldots+c_ nx_ n|_ v$$.
Suppose that $$S$$ is a finite set of valuations of $$K$$ containing the archimedean valuations and write $$U_ s$$ for the set of all $$S$$-units, i.e. of all $$\alpha \in K^*$$ such that $$| \alpha |_ v=0$$ for $$v\not\in S$$. Many authors are concerned with the “qualitative theory” of the $$S$$-unit equation $$c_ 0x_ 0+\ldots+c_ nx_ n=0$$, i.e. the number of possible solutions.
In the paper under review, the author is concerned with the “quantitative theory” to obtain results which could be seen as a refinement of Evertse’s bound $$N({\mathfrak x})>c H({\mathfrak x})^{1+\varepsilon}$$. His techniques are related to the Hardy-Littlewood method, and use the theorem of Baker on linear forms in logarithms.

##### MSC:
 11G50 Heights 11J86 Linear forms in logarithms; Baker’s method 11D57 Multiplicative and norm form equations 11R27 Units and factorization
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##### References:
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