# About algebraic Puiseux series in several variables

## TL;DR

This paper investigates the algebraic properties of multivariable Puiseux series, establishing criteria for algebraicity based on coefficients and providing explicit formulas for series coefficients in terms of polynomials.

## Contribution

It generalizes previous results to several variables, offering explicit universal polynomial formulas and closed-form expressions for coefficients.

## Key findings

- Algebraicity determined by finite universal polynomial formulas
- Explicit formulas for coefficients in terms of vanishing polynomials
- Generalization of single-variable results to multivariable case

## Abstract

We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [HM15]. We show that the algebraicity of such a series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of a bounded initial part of the series.