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 $f_{\lambda }^m({\alpha }_1(\lambda ))=f_{\lambda }^n({\alpha }_2(\lambda ))=\beta (\lambda )$.