Unirational Differential Curves and Differential Rational Parametrizations

TL;DR

This paper explores the properties of unirational differential curves and develops algorithms for their parametrization and implicitization using differential resultants, advancing the understanding of differential algebraic geometry.

Contribution

It introduces methods for analyzing proper differential rational parametrizations and provides an algorithm to determine unirationality and compute parametrizations for linear differential curves.

Findings

Proper differential rational parametrizations characterized

Implicitization via differential resultants solved

Algorithm for linear differential curves developed

Abstract

In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper differential rational parametrization for the differential curve.