The combinatorial Jacobian conjecture

Fix an integer $d\geq 3$. For each $n\geq 1$, let $H=(H_{i,\alpha}:i\in[n],\lvert\alpha\rvert=d)$ be a collection of complex numbers, and let $\Psi_{i,j}^{\alpha}(H)$ and $\Phi_i^{\alpha}(H)$ denote the combinatorial quantities defined in the paper. The combinatorial Jacobian conjecture. There exists $d\geq 3$ such that, for every $n\geq 1$, if

$$\Psi_{i,j}^{\alpha}(H)=0$$

for all $i,j\in[n]$ and $\alpha\in\mathbb{Z}_{\geq 0}^n$ with $\lvert\alpha\rvert=(d-1)n$, then, for each $i\in[n]$, $\Phi_i^{\alpha}(H)=0$ for all but finitely many $\alpha$. This is an equivalent combinatorial formulation of the Jacobian conjecture, encoding nilpotency conditions and polynomiality of the inverse through weighted sums over labelled trees.