The Valqui–Guccione–Guccione criterion for the Jacobian conjecture

Let $a,b\in\mathbb{Z}_{>0}$ satisfy $a\nmid b$ and $b\nmid a$. Let $\lambda_i\in\mathbb{C}$ for $i\in[0,a+b-2]$, with $\lambda_0=1$. Suppose that $F,G\in\mathcal{R}=\mathbb{C}[x,y]$ and $C,P\in\mathbb{C}[y]((x^{-1}))$ satisfy the five stated conditions: $C$ has the prescribed Laurent expansion, $\deg(C)=1$ and $\deg(P)\leq2-a$, the specified leading part of $P$ when equality holds, $C^a=G$ and $F=\sum_{i=0}^{a+b-2}\lambda_iC^{b-i}+P$, and $[F,G]\in\mathbb{C}$.

The Valqui–Guccione–Guccione criterion. Then

$$[F,G]=0.$$

The paper identifies this as a slight reformulation of a conjecture equivalent to the Jacobian conjecture, citing Valqui, Guccione, and Guccione. The source supplies no resolution status.