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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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