On the algebraicity of Puiseux series

TL;DR

This paper investigates the algebraic properties of Puiseux series, establishing explicit polynomial criteria for their algebraicity and providing formulas for their coefficients based on associated polynomials.

Contribution

It introduces finite polynomial conditions to determine the algebraicity of Puiseux series and offers closed-form formulas for their coefficients from given polynomials.

Findings

Algebraicity of Puiseux series is characterized by explicit polynomial conditions.

Finite criteria determine algebraicity for bounded degree series.

Coefficients can be explicitly computed from polynomials and initial terms.

Abstract

We deal with the algebraicity of a Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degree is determined by a finite number of explicit polynomial formulae. 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 an initial part of the series.