Computing Hypercircles by Moving Hyperplanes

TL;DR

This paper introduces a new algorithm for computing hypercircles associated with rational curves, enabling determination of the curve's field of definition and its minimal field extension, without complex normal closure computations.

Contribution

The paper presents a novel algorithm that efficiently computes hypercircles and determines the minimal field of definition for rational curves over characteristic zero fields.

Findings

Algorithm successfully computes hypercircles from parametrizations.

It determines whether a curve is defined over a given field.

It identifies the minimal field extension containing the curve.

Abstract

Let K be a field of characteristic zero, alpha algebraic of degree n over K. Given a proper parametrization psi of a rational curve C, we present a new algorithm to compute the hypercircle associated to the parametrization psi. As a consequence, we can decide if the curve C is defined over K and, if not, to compute the minimum field of definition of C containing K. The algorithm exploits the conjugate curves of C but avoids computation in the normal closure of K(alpha) over K.