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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

Citations:

Zbl 0498.12030
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