MR2232636

Boussaf, Kamal(F-CLEF2-LPM)

Uniqueness for $p$-adic meromorphic products. (English summary)

Bull. Belg. Math. Soc. Simon Stevin 9 (2002), suppl., 11--23.

32P05 (32H04)

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In the paper under review the author looks for bi-unique range sets (bi-urs) for the family of unbounded meromorphic products on an open disk. More precisely, let $K$ be a complete ultrametric algebraically closed field of characteristic zero, and let $\scr{M}(K)$ be the field of meromorphic functions in $K$. Denote by $\scr{MP}_{u}(K,R)$ the subset of meromorphic products admitting an irreducible form $\prod_{n=0}^{\infty}\frac{x-a_n}{x-b_n}$ such that $\prod_{n=0, b_n\neq 0}^{\infty}\frac{|b_n|}{R}=0$. The main result in the paper under review implies that for every $n\geq5$, there exist sets $S$ of $n$ elements in $K$ such that $(S,\{\infty\})$ is a bi-urs for $\scr{MP}_u(K,R)$.

Earlier, A. Boutabaa and A. Escassut proved that for every $n\geq5$, there exist sets $S$ of $n$ elements in $K$ such that $(S,\{w\})$ is a bi-urs for $\scr{M}(K)$. H. H. Khoi and T. T. H. An showed the existence of bi-urs for $\scr{M}(K)$ of the form $(\{a_1,a_2,a_3,a_4\},\{\infty\})$.

Reviewed by Karl-Olof Lindahl