## Produits méromorphes.(French)Zbl 0564.12024

Let $$K$$ be a complete ultrametric algebraically closed field, let $$b_n$$ be a sequence in $$K$$ such that $$| b_n| <R=\lim_{n\to \infty}| b_n|$$ and for each $$\rho >0$$ let $$\Lambda_{\rho}=\{x\in K \mid | x-b_n| \geq \rho \forall n\in \mathbb{N}\}$$. Let $$a_n$$ be a sequence in $$K$$ such that $$\lim_{n\to \infty}(a_n-b_n)=0$$ and let $$\pi (x)=\prod_{n\in \mathbb{N}}(x-a_n)/(x-b_n)$$. The author shows that $$\pi(x)$$ is an analytic element on $$\Lambda_{\rho}$$ and that it also lies in the quotient field of the Taylor series convergent for $$| x| <R$$.
For every $$\lambda\ne -1$$, $$\pi(x)+\lambda$$ can be factorized in the form $$\lambda\prod_{n\in\mathbb{N}}(x-c_n)/(x-b_n)$$ with $$c_n$$ a new sequence such that $$\lim_{n\to \infty}(b_n-c_n)=0$$, and it is quasi-invertible in $$H(\Lambda_{\rho})$$ (the algebra of the analytic elements on $$\Lambda_{\rho})$$.
For $$\lambda =-1$$, $$\pi(x)-1$$ is sometimes quasi-invertible, then it has a form $(1/Q(x)) \prod^{\infty}_{n=q}(x-c_n)/(x-b_n)$ with $$Q$$ a polynomial whose zeros are some $$b_n$$. But if $$\Lambda_{\rho}$$ has an increasing T-filter the author proves there do exist products $$\pi(x)$$ such that $$\lim_{| x| \to R}\pi(x)=1$$. They are not quasi- invertible in $$H(\Lambda_{\rho})$$ and they have no factorization looking like the previous ones. They are called collapsing products. That requires results obtained by the author and the reviewer in “T-suites idempotentes” [ibid. 106, 289–303 (1982; Zbl 0498.12030)]. That also proves the inverse of a nonbounded Taylor series convergent for $$| x| <R$$ may belong to $$H(\Lambda_{\rho})$$.

### MSC:

 12J25 Non-Archimedean valued fields 26E30 Non-Archimedean analysis 30G06 Non-Archimedean function theory

Zbl 0498.12030