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Pseudo-convergent sequences and Prüfer domains of integer-valued polynomials. (English) Zbl 1345.13013

Is \(D\) is an integral domain with quotient field \(K\) and \(S\subset D\), then denote by \(\mathrm{Int}(S,D)\) the set of polynomials \(f\in K[X]\) satisfying \(f(S)\subset D\). If \(\mathrm{Int}(S,D)\) is a Prüfer domain, then \(D\) is also Prüfer, and the converse holds in case of finite \(S\), as shown in [Proc. R. Ir. Acad., Sect. A 85, 177–184 (1985; Zbl 0596.13017)] by D. L. McQuillan.
In case when \(D\) a valuation domain and \(S\) is precompact, i.e., it has a compact completion, then it has been established by P.-J. Cahen et al. [J. Korean Math. Soc. 38, No. 5, 915–935 (2001; Zbl 1010.13011)] that \(\mathrm{Int}(S,D)\) is Prüfer, and the precompact condition is also necessary if the valuation is discrete.
The authors study the case when \(D\) is the valuation domain of a field with a rank one valuation and \(S =\{a_n\}\subset D\) is a pseudo-convergent sequence, i.e. for \(i>j>k\) one has \(v(a_i-a_j)>v(a_j-a_k)\), as defined by A. Ostrowski [Math. Z. 39, 269–320, 321–361, 361–404 (1934; JFM 60.0899.01)]. In Theorem 5.2 they give a necessary and sufficient condition for \(\mathrm{Int}(S,D)\) to be Prüfer, from which it follows that the precompact condition is not necessary.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
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References:

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