The theory of stratification relative to a Newton polyhedron. (Q1890154)

A stratification of a variety \(V\) is an expression of \(V\) as the disjoint union of a locally finite sets of connected analytic manifolds, called strata, such that the boundary of each stratum is the union of a set of lower-dimensional strata. The most important notion in stratification theory is the regularity condition between strata. In this paper, the author investigates regularity conditions relative to a fixed Newton filtration in the context of stratification theory. First, the author shows a characterization of the (\(c\))-regularity condition defined by \textit{K. Bekka} [Lect. Notes Math. 1462, 42--62 (1991; Zbl 0733.58003)]. Next, the author presents a criterion for regularity conditions in terms of the defining equations of the strata and, after introducing a pseudo-metric adapted to the Newton polyhedron, obtains versions relative to the Newton filtration of the Fukui-Paunescu theorem [\textit{T. Fukui} and \textit{L. Paunescu}, Can. J. Math. 53, No. 1, 73--97 (2001; Zbl 0983.32006)]. In this approach it is possible to consider the version relative to a Newton filtration of the \((w)\)-regularity condition. It is shown that this condition implies \((c)\)-regularity condition. Finally, the author shows that the Demon-Gaffney condition [\textit{J. Damon} and \textit{T. Gaffney}, Invent. Math. 72, 335--358 (1983; Zbl 0519.58021)] implying the topological triviality of an analytic deformation of an analytic function can be expressed also in terms of Newton filtrations and proves that it implies \((c)\)-regularity condition.