Resumen de Characterization of the collapsing meromorphic products

Alain Escassut, Marie-Claude Sarmant

• Let $K$ be an algebraically closed complete ultrametric field. Let $a\in K$, $r>0$. We consider a meromorphic product $F(x)= \displaystyle\prod_{n \in \Bbb N} \frac{x -a_n}{x - b_n}$, where $(a_n)_{n\in \Bbb N}$, $(b_n)_{n\in \Bbb N}$ are sequences satisfying $|b_n-a|0$. We prove that if $K$ has characteristic zero, then $F$ is collapsing if and only if $\displaystyle\sum_{n=0}^\infty (a_n)^j-(b_n)^j=0$ for every $j\in \Bbb N$. Moreover, if $K$ has characteristic $\ne 0$, then there exists a meromorphic product $f$ of the form $\displaystyle\prod_{n \in \Bbb N} \frac{x-c_n}{x - e_n}$ such that $F(x)=(f(x))^p$ whenever $x\in \{ x\in K |\ |x-a|\geq r\}$ if and only if $\displaystyle\sum_{n=0}^\infty (a_n)^j-(b_n)^j=0$ for every $j\in \Bbb N$.