Newton trees for ideals in two variables and applications

TL;DR

This paper introduces Newton trees and an algorithm to analyze arbitrary ideals in two variables, enabling the computation of invariants like multiplicity and valuations through combinatorial structures.

Contribution

It develops a Newton algorithm and Newton tree framework to generalize the analysis of ideals in two variables beyond special cases.

Findings

Newton trees encode ideal data combinatorially.

Rees valuations correspond to dicritical vertices of the Newton tree.

Hilbert-Samuel multiplicity is computable from the Newton tree.

Abstract

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.