Abstract: Let $f,g: \mathbb{P}^1 \to \mathbb{P}^1$ be two rational maps of degree $d \geq 2$. DeMarco, Krieger and Ye have conjectured that there exists an uniform bound $B = B(d)$, depending only on the degree $d$, such that the number of common preperiodic points between $f$ and $g$ is either $\leq B$ or infinite. In this talk, we will show that a statistical version of this conjecture for polynomials is true. We will also explore what the smallest possible value of $h_f(x) + h_g(x)$ can be for a generic pair of polynomials, where $h_f$ and $h_g$ denotes the canonical height of $f$ and $g$ respectively.
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