Shared dynamically small points for polynomials on average

Given two rational maps f,g: P¹ → P¹ of degree d over C , DeMarco, Krieger, and Ye [Common preperiodic points for quadratic polynomials. J. Mod. Dyn. 18 (2022), 363–413] have conjectured that there should be a uniform bound $B = B(d)> 0$ such that either they have at most B common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree d ≥ 6 with rational coefficients. We also investigate the quantity x ∈ Q̄ ( hf(x) + hg(x) ) for a generic pair of polynomials and prove both lower and upper bounds for it.