Remark 4

Suppose that $p + q \geq 1$. By III, p. 532, formula (28), the image of $\varphi$ contains the constant polynomial $\operatorname{res}_{p,q}(f, g)$. Hence there exist a pair of polynomials $(u, v)$, with $u \in S_q, v \in S_p$ such that

whence This pair $(u, v)$ is unique when $\operatorname{res}_{p,q}(f, g)$ is cancellable in $A$: for then $\varphi$ is injective (III, p. 524, Prop. 3).