Article ID Journal Published Year Pages File Type
4597612 Journal of Pure and Applied Algebra 2010 15 Pages PDF
Abstract
Suppose that k is an arbitrary field. Let k[[x1,…,xn]] 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¯[[x1,…,xn]]. We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k[[x1,…,xn]]. 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.
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Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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