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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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