Let L be a field of positive characteristic p with a fixed algebraic closure \(\overline{L}\) , and let \(\alpha _1,\alpha _2,\beta \in L\) . For an integer \(d\ge 2\) , we consider the family of polynomials \(f_{\lambda }(z):= z^d+\lambda \) , parameterized by \(\lambda \in \overline{L}\) . Define \(C(\alpha _1,\alpha _2;\beta )\) to be the set of all \(\lambda \in \overline{L}\) for which there exist \(m,n\in {\mathbb {N}}\) such that \(f_{\lambda }^m(\alpha _1)=f_{\lambda }^n(\alpha _2)=\beta \) . In other words, \(C(\alpha _1,\alpha _2;\beta )\) consists of all \(\lambda \in \overline{L}\) with the property that the orbit of \(\alpha _1\) collides with the orbit of \(\alpha _2\) under the same polynomial \(f_{\lambda }\) precisely at the point \(\beta \) . Assuming \(\alpha _1,\alpha _2,\beta \) are not all contained in a finite subfield of L, we provide explicit necessary and sufficient conditions under which \(C(\alpha _1,\alpha _2;\beta )\) is infinite. We also discuss the remaining case where \(\alpha _1,\alpha _2,\beta \in \overline{\mathbb {F}}_p\) and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic 0. Working in characteristic p adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when d is a power of p.
