Collision of Orbits for Families of Polynomials Defined over Number Fields

Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_\lambda(z):=z^d+\sum_{i=0}^{d-2} c_i(\lambda)\cdot z^i$ parameterized by $\lambda\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$. Also, let $\alpha_1(t),\alpha_2(t),\beta(t)\in\bar{\mathbb{Q}}[t]$, where $\alpha_i(t)$ is not preperiodic under the action of $f_t(z)$ for each $i=1,2$. Under some natural hypotheses, we obtain precise necessary and sufficient conditions for which there exist infinitely many $\lambda\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $\lambda$), we have that $f_\lambda^m(\alpha_1(\lambda))=f_\lambda^n(\alpha_2(\lambda))=\beta(\lambda)$.