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Thue’s theorem in positive characteristic. (English) Zbl 0844.11046
Let \(K\) be any field of positive characteristic \(p\). For any element \(f\in K ((T^{-1}))\), we define \(\deg f\) by \(\deg f= n_0\) if \(f= \sum^{n_0}_{n=-\infty} a_n T^n\), where the \(a_n\) are coefficients in \(K\) with \(a_{n_0} \neq 0\), and \(\deg 0= -\infty\). We define then an absolute value on \(K(( T^{-1}))\) by \(|f|= |T|^{\deg f}\), where \(|T|> 1\). The valued field \(K(( T^{-1} ))\) can be considered as the completion of the field of rational fractions \(K(T)\).
Let \(\alpha\) be an algebraic element of \(K(( T^{-1} ))\) of degree \(n>1\) over \(K(T)\). We prove here that a result exactly similar to Thue’s theorem holds for \(\alpha\) (i.e. \(|Q\alpha- P|\gg|Q|^{-( [n/2 ]+ \varepsilon)}\) for each \(\varepsilon> 0\) and for every pair \((P, Q)\) of polynomials in \(K[T]\), \(Q\neq 0\)), if \(\alpha\) satisfies no equation \(\alpha (A \alpha^{p^s}+ B)/(C \alpha^{p^s}+ D)\), where \(n\) is a positive integer, and \(A\), \(B\), \(C\), and \(D\), polynomials in \(K[T]\), not all zero.

MSC:
11J25 Diophantine inequalities
11D75 Diophantine inequalities
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