Puiseux power series solutions for systems of equations

TL;DR

This paper introduces an algorithm for computing multivariate Puiseux series expansions of solutions to algebraic systems at singular points, extending Newton's method via tropical geometry techniques.

Contribution

It presents a novel algorithm that generalizes Newton's method using tropical varieties to handle multivariate Puiseux series solutions.

Findings

Algorithm successfully computes series expansions at singular points.

Extension of Newton's method to multivariate cases.

Deduces properties of tropical varieties for quasi-ordinary singularities.

Abstract

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.