The Newton-Puiseux algorithm and effective algebraic series

TL;DR

This paper presents methods to encode algebraic series efficiently, perform effective arithmetic, and analyze their support using the Newton-Puiseux algorithm, enabling identification of polynomial and rational solutions.

Contribution

It introduces a finite data encoding for algebraic series and develops effective algorithms for arithmetic, equality testing, and solution identification based on the Newton-Puiseux algorithm.

Findings

Finite data encoding of algebraic series

Effective arithmetic and equality testing methods

Algorithms for identifying polynomial and rational solutions

Abstract

We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series. Furthermore, we discuss how to derive information about the support of an algebraic series. Based thereon, we show how to identify the polynomial and rational solutions of a polynomial equation.