Uniformization of multivariate power series


In this paper, we describe an algorithm for the “uniformization” of a multivariate power series. Let be the field of “grid-based power series” over a sufficiently large non archimedean “monomial group” (or value group) , such as with the lexicographical ordering on . We interpret power series as functions . On certain “regions” of the space , it may happen that the valuation of can be read off from the valuations of the . In that case, is said to be “uniform” on . We will describe an algorithm for cutting into a finite number of regions, each on which is uniform for a suitable change of coordinates, which preserves the elimination ordering on . The algorithm can probably be seen as an effective counterpart of local uniformization in the sense of Zariski, even though this connection remains to be established in detail.

Keywords: algorithm, power series, grid-based power series, local uniformization, Newton polygon method, desingularization

A.M.S. subject classification: 14B05, 68W30, 14E15

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