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


11J70 Continued fractions and generalizations
11J61 Approximation in non-Archimedean valuations
11D88 \(p\)-adic and power series fields
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