... = 0 be the implicit equation of (s(x), t(x)). Let (s(x), t(x)) be a proper parametrization of F (s, t) = 0. By the relation between two rational parametrizations of the same algebraic curve (see e.g =-=[SWPD08]-=-, Lemma 4.17), there exists a rational function T (x) such that { s(x) = s(T (x)) (8) t(x) = t(T (x)). It follows from (7) that { s ′ (T (x)) · T ′ (x) = P (s(T (x)), t(T (x))) t ′ (T (x)) · T ′ (x) =...

...t) and the coefficients of G(s, t). In fact, G(s, t) is uniquely defined by the quotient of the division FsP +FtQ by F . Thus we only need to solve a system of equations on the coefficients of F (see =-=[Man93]-=-). This observation makes the computation of invariant algebraic curves more effectively because one need not to involve more equations and variables from the coefficients of G(s, t). It is known that...

...s differential equation with unknow T (x). Moreover, this differential equation is of degree 1 with respect to T ′(x). Therefore, its rational solutions are linear fractional transformations (see e.g =-=[FG06]-=-). Hence (s(x), t(x)) is a proper rational solution. (2). It follows from the above construction immediately. ✷ Algorithm 1. Input: P (s, t), Q(s, t), F (s, t) such that FsP + FtQ = F G for some G. Ou...