Characterization of the collapsing meromorphic products (Q677833)

Let \(K\) be an algebraically closed, complete non-archimedean valued field. Let \(a\in K\), \(r>0\). For sequences \((a_n)\), \((b_n)\) in \(K\) for which \(| b_n-a |<r\), \(\lim_{n\to\infty} | b_n-a |=r\), \(\min \{| b_m-b_n|: m\neq n\}>0\), \(\lim_{n\to\infty} (b_n-a_n)=0\), we define the meromorphic product \[ F(x)= \lim_{m\to\infty} \prod^m_{n=0} {x-a_n\over x-b_n} \text{ for }x\in K\setminus \{b_0,b_1, \dots\}. \] The authors show that \(F'(x)=0\) for all \(x\in K\), \(| x-a|\geq r\) iff (*): \(\sum_n(a^j_n-b^j_n) =0\) for each \(j\in\mathbb{N}\). As a corollary they obtain if \(\text{char} K=0\) that (*) holds iff \(F\) is collapsing (i.e. \(\lim_{| x-a |\to r} F(x)\) exists), whereas if \(\text{char} K=p\neq 0\) one has (*) iff \(F(x)=f(x)^p\) where \(f(x)=\prod_n{x-c_n\over x-e_n}\) is a meromorphic product is the style of above.