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Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $\gamma \mapsto ord\left(Res\left({\phi }^{\gamma }\right)\right)$ factors through a function $ordRe{s}_{\phi }\left(·\right)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P{¹}_K$, or on a segment, and the minimal resultant locus is contained in the tree in $P{¹}_K$ spanned by the fixed points and poles...

Given a rational function $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$. The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by...