Newton trees for ideals in two variables and applications

We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For instance when the ideal is of finite codimension, invariants like integral closure and Hilbert-Samuel multiplicity were already combinatorially determined in the very special cases of monomial or non degenerate ideals, using the Newton polygon of the ideal. With our approach, we can generalize these results to arbitrary ideals. In particular the Rees valuations of the ideal will correspond to the so-called dicritical vertices of the tree, and its Hilbert-Samuel multiplicity has a nice and easily computable description in terms of the tree.