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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.

11G50 Heights
11J86 Linear forms in logarithms; Baker’s method
11D57 Multiplicative and norm form equations
11R27 Units and factorization
Full Text: Numdam EuDML
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