Encoding algebraic power series

TL;DR

This paper presents a method to encode algebraic power series using polynomial vectors, enabling finite algorithms for division and manipulation of these series based on their codes.

Contribution

It introduces a novel encoding of algebraic power series via polynomial vectors and demonstrates how to perform division operations directly on these codes.

Findings

Finite algorithms for division of algebraic series are developed.

Algebraic series can be completely described by polynomial codes.

Division operations on codes preserve algebraic properties.

Abstract

Algebraic power series are formal power series which satisfy a univariate polynomial equation over the polynomial ring in n variables. This relation determines the series only up to conjugacy. Via the Artin-Mazur theorem and the implicit function theorem it is possible to describe algebraic series completely by a vector of polynomials in n+p variables. This vector will be the code of the series. In the paper, it is then shown how to manipulate algebraic series through their code. In particular, the Weierstrass division and the Grauert-Hironaka-Galligo division will be performed on the level of codes, thus providing a finite algorithm to compute the quotients and the remainder of the division.