Characteristic polyhedra of singularities without completion
Let \((R, \mathfrak{m})\) be a regular local ring and \(y, u_1, \ldots, u_d\) be a regular system of parameters. Let \(f\in \mathfrak{m}\) and \(f\not\in \langle u_1, \ldots, u_d\rangle\). Hironaka associated a polyhedron \(\Delta(f; u_1, \ldots, u_d;y)\subset \mathbb{R}^d_{\geq 0}\) and proved that there exists \(z\in\widehat{R}\) such that \(\{z, u_1, \ldots, u_d\}\) is a regular system of parameters of \(\widehat{R}\) and \[ \Delta(f; u_1, \ldots, u_d; z)=\underset{(\widehat{y}, u_1, \ldots, u_d)}{\bigcap} \Delta(f; u_1, \ldots, u_d, \widehat{y}) \] where the intersection is over all regular systems of parameters of \(\widehat{R}\) of the form \((\widehat{y}, u_1, \ldots, u_d)\), (c.f. \textit{H. Hironaka} [J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)]). It is proved that \(z\) can be chosen in \(R\) in case \(R\) is a \(G\)--ring (i.e. the formal fibres are geometrically regular).
characteristic polyhedron of a singularity
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