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Continued fractions of algebraic power series in characteristic 2. (English) Zbl 0312.10024
Let $$K=\mathbb F_2 ((x^{-1}))$$ be the field of formal power series in $$x^{-1}$$ over $$F= \mathbb F_2$$, the field with two elements. There is a continued fraction theory for $$K$$, analogous to that for real numbers, with polynomials in $$x$$ playing the role of the integers. The following theorems are shown:
Theorem 1. If $$f\in K$$ satisfies $$f^3+p^{-1}f+1=0$$ where $$p\in F[x]$$ and $$\deg p\geq 1$$, then $$f$$ has bounded partial quotients.
Theorem 2. Let $$n\geq 0$$, and let $$P,Q\in F[x]$$ be such that $$P-Q^{2^n}\neq 0,1$$. Then $f^{2^n+1}+Qf^{2^n}+Pf+QP+1=0$ has a unique root $$f$$ in $$K$$, and we have $f=[Q; P+Q^{2^n}, P^{2^n}+Q^{2^{2n}}, P^{2^{2n}}+Q^{2^{3n}},\ldots].$ It follows from Theorem 2 that the “Diophantine equation” $P^{2^n+1}+Qp^{2^n}q+(QP+1)q^{2^n+1}=1$ has infinitely many polynomial solutions $$p,q\in F[x]$$, where $$P$$ and $$Q$$ are fixed polynomials in $$F[x]$$ with $$P+Q^{2^n+1}\neq 0,1$$. Moreover, all solutions of this equation are classified. Furthermore, the continued fraction expressions for a second class of algebraics with unbounded partial quotients is given, viz. the algebraics $$(P/(P+1))^{1/(2^n-1)}\in K$$, where $$P\in F[x]$$, $$P\neq 0,1$$. This yields a classification of the infinitely many solutions $$p,q\in F[x]$$ of $(P+1)p^{2^n-1}+Pq^{2^n-1}=1.$
Reviewer: Leonard E. Baum

##### MSC:
 11J70 Continued fractions and generalizations 11J61 Approximation in non-Archimedean valuations 11D88 $$p$$-adic and power series fields
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