Dynamic Newton-Puiseux Theorem

TL;DR

This paper presents a constructive, dynamic approach to the Newton-Puiseux theorem for algebraic curves, allowing computation of Puiseux expansions without requiring an algebraically closed base field or polynomial factorization.

Contribution

It introduces a novel, constructive method based on Abhyankar's classical proof, enabling dynamic evaluation of algebraic numbers and expansions over regular algebras.

Findings

Allows computation of Puiseux expansions over non-closed fields

Eliminates the need for polynomial factorization over the base field

Provides a constructive, dynamic framework for algebraic curve analysis

Abstract

A constructive version of Newton-Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regular algebras over the base field and the expansions are given as formal power series over these algebras.