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A characterization of rational elements by Lüroth-type series expansions in the \(p\)-adic number field and in the field of Laurent series over a finite field. (English) Zbl 1095.11040
If \(|\,.\,|_p\) denotes the \(p\)-adic valuation of \(\mathbb{Q}_p\), normalized by \(|p|_p=p^{-1}\), the order \(\nu(A)\in \mathbb{Z}\) of \(A\in \mathbb{Q}_p^\times\) is defined by \(|A|_p=p^{-\nu(A)}\), and \(\nu(0):=+\infty\). Furthermore, every \(A\in \mathbb{Q}_p^\times\) has a unique series representation \(A=\sum_{n\geq \nu(A)}c_np^n\) with all \(c_n\in \{0,\dots,p-1\}\) and \(c_{\nu(A)}\neq 0\); then the rational number \(\langle A\rangle:=\sum_{n=\nu(A)}^0 c_np^n\) is called the fractional part of \(A\), and \(S_p\) denotes the set of all possible fractional parts.
With these notations, A. Knopfmacher and J. Knopfmacher [J. Number Theory 32, 297–306 (1989; Zbl 0683.10030)] proved the following \(p\)-adic Lüroth-type theorem. Every \(A\in \mathbb{Q}_p\) has a finite or (\(p\)-convergent) unique series expansion of the form \(A=a_0+1/a_1+\sum_{n>1}1/(a_1(a_1-1)\cdots a_{n-1}(a_{n-1}-1)a_n)\), where all digits \(a_n\) are in \(S_p,\, a_0=\langle A\rangle\), and \(\nu(a_n)\leq-1\) for each \(n\geq1\). If the sequence \((a_n)\) is finite or eventually periodic, then \(A\) is rational.
At the end of their paper, and again in [J. Number Theory 41, 131–145 (1992; Zbl 0756.11021)], the Knopfmachers asked if each infinite Lüroth expansion representing a rational number must be periodic. It is the main aim of the present paper to answer this question in the affirmative. In addition, a similar characterization is established in the (technically simpler case of) the field of Laurent series over a finite field.

MSC:
11J72 Irrationality; linear independence over a field
11J61 Approximation in non-Archimedean valuations
11A67 Other number representations
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