Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[ S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \existsλ\