The Qth-power algorithm in characteristic 0

TL;DR

This paper extends the Qth-power algorithm, originally for finite fields, to characteristic zero by combining modular computations over small primes with the Chinese remainder theorem and Euclidean algorithm, enabling reconstruction over the rationals.

Contribution

It introduces a method to lift the Qth-power algorithm from finite fields to characteristic zero using modular techniques and rational reconstruction.

Findings

Successfully reconstructs integral closures over $\,\mathbb{Q}$ from modular data.

Provides a practical approach for characteristic zero computations in algebraic geometry.

Demonstrates the effectiveness of combining Chinese remainder theorem with Euclidean algorithm.

Abstract

The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=\mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $\mathbf{F}:=\mathbf{Z}_q$ of $q$ elements. Here it is shown how to use this for several small primes $q$ to reconstruct similar integral closures over the rationals $\mathbf{Q}$ using the Chinese remainder theorem to piece together presentations in different positive characteristics, and the extended Euclidean algorithm to reconstruct rational fractions to lift these to presentations over $\mathbf{Q}$.