Abstract:
 Let $k$ be a field and let $k^s$ be the separable closure of $k$ in the algebraic closure kc of $k$. An algebraic zero divisor (P) on the classical projective line $\mathbb{P}^1(k^c)$ over kc, which is the zeros divisor on $\mathbb{P}^1(k^c)$ of a homogeneous polynomial $\text{P}$ of two variables over $k$, generalizes both the Galois conjugacy class of a point in $\mathbb{A}(k^c)$ and the roots divisor of the equation $f=a$ on $\mathbb{P}^1(k^c)$ for distinct rational functions $f$, $a\in k(z)$ satisfying min{deg $f$, deg$a$} > 0. The mass of (P)×(P) on the diagonal of $\mathbb{P}^1(k^c)$x$\mathbb{P}^1(k^c)$, called the diagonal of (P), play a nontrivial role in the studying such an algebraic zero divisor (P) more general than Galois conjugacy classes. If $k$ is a product formula field, i.e., a field equipped with adelic places satisfying the product formula, then we can also associate to the algebraic divisor (P) a height $h_g$((P)) with respect to an adelic family g of normalized continuous weights (or potentials) on the Berkovich projective lines $\text{P}^1(\mathbb{C}_v)$ for all places $v$ on $k$, in a similar way to define the classical heights of Galois conjugacy classes. In this talk, we give a generalization of the adelic equidistribution theorem due to BakerRumely, FavreRiveraLetelier, and ChambertLoir and its quantitative version due to FavreRiveraLeteier for a sequence of Galois conjugacy classes on $\mathbb{P}^1(k_s)$ having small heights. This generalization is suitable for studying both the dynamics and the value distribution of the iteration of a rational function $f\in\mathbb{Q}_c(z)$ of degree more than one since the roots divisor $f^n$=a for a rational function $a\in\mathbb{Q}_c(z)$ and $n\in\mathbb{N}$ large enough is not necessarily a Galois conjugacy classes in $\mathbb{A}(kc)$ but always an algebraic zeros divisor on $\mathbb{P}^1(k^c)$ for a fixed number field $k$.
