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• W2144391724 abstract "Suppose that k is an arbitrary field. Let k [ [ x 1 , … , x n ] ] be the ring of formal power series in n variables with coefficients in k . Let k ¯ be the algebraic closure of k and σ ∈ k ¯ [ [ x 1 , … , x n ] ] . We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k [ [ x 1 , … , x n ] ] . We also characterize valuation rings V dominating an excellent Noetherian local domain R of dimension 2, and such that the rank increases after passing to the completion of a birational extension of R . This is a generalization of the characterization given by M. Spivakovsky [Valuations in function fields of surfaces, Amer. J. Math. 112 (1990) 107–156] in the case when the residue field of R is algebraically closed." @default.
• W2144391724 created "2016-06-24" @default.
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• W2144391724 date "2008-08-01" @default.
• W2144391724 modified "2023-09-29" @default.
• W2144391724 title "Algebraic series and valuation rings over nonclosed fields" @default.
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• W2144391724 doi "https://doi.org/10.1016/j.jpaa.2007.11.013" @default.
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