Computing the \textbf{V}-saturation of finitely-generated submodules of \(\mathbf{V} [X]^m\) where \textbf{V} is a valuation domain
The article describes an algorithm to compute the \(\mathbf{V}\)-saturation of a finitely generated submodule of \(\mathbf{V}[X]^n\), where \(V\) is a saturation domain, breaking it down into several specific subalgorithms and analyzing its complexity. Here, \(X\) denotes only one variable, so the algorithm is for the univariate case; the authors nod to a generalization for the multivariate case of Krull dimension 1; this result applies to the univariate case of arbitrary Krull dimension. The article proves (a) correctness if the algorithm terminates; (b) termination after a specified number of operations; and (c) complexity bounds. Two applications are given: a test for an analogue of ideal membership called saturation membership, and the computation of syzygies over the polynomial ring of a valuation domain, proving the special case that in the case of a Prüfer domain \(\mathbf{R}\) the \(\mathbf{R}\)-saturation of a finitely-generated submodule of \(\mathbf{R}[X]^m\) is finitely generated and the polynomial ring \(\mathbf{R}[X]\) is coherent.
scientific article; zbMATH DE number 6473752