$D$-resultant for rational functions
Jaime Gutiérrez, Rosario Rubio, Jie-Tai Yu · Proceedings of the American Mathematical Society · 2002
In this paper we introduce the $D$-resultant of two rational functions $f(t),g(t) \in \mathbb {K}(t)$ and show how it can be used to decide if $\mathbb {K}(f(t),g(t))=\mathbb {K}(t)$ or if $\mathbb {K}[t]\subset \mathbb {K}[f(t),g(t)]$ and to find the singularities of the parametric algebraic curve define by $X=f(t), Y=g(t)$. In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which has its own interest.