Uniformization of multivariate power series

In this paper, we describe an algorithm for the “uniformization” of a multivariate power series. Let K[[T]] be the field of “grid-based power series” over a sufficiently large non archimedean “monomial group” (or value group) T, such as T = {t1 α1 tnα: α1, , αn ∈ R} with the lexicographical ordering on α1, , αn. We interpret power series f ∈K[[x1, , xn]] as functions K[[T]]→K[[T]]. On certain “regions” R of the space K[[T]], it may happen that the valuation of f can be read off from the valuations of the xi. In that case, f is said to be “uniform” on R. We will describe an algorithm for cutting K[[T]] into a finite number of regions, each on which f is uniform for a suitable change of coordinates, which preserves the elimination ordering on x1, , xn. 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.