Take holomorphic $f(x,y)$, $g(x,y)$. A polar arc is a Puiseux root, $x = \gamma(y)$, of the Jacobian $J = f_y g_x - f_x g_y$, but not one of $f \cdot g$. We define the tree, $T(f,g)$, using the contact orders of the roots of $f \cdot g$, describe how polar arcs climb, and leave, the tree, and how to factor $J$ in $\mathbf{C}\{x,y\}$. When collinear points/bars exist, the way the $\gamma$'s leave the tree is not an invariant.

We prove under certain natural conditions the finiteness of the number of isomorphism classes of Zariski dense subgroups in semisimple groups with isomorphic $p$-adic closures.

We consider the Hecke $L$-function $L(s,\lambda^m)$ of the imaginary quadratic field $\mathbf{Q}(i)$ with the $m$-th Grossencharacter $\lambda^m$. We obtain the universality property of $L(s,\lambda^m)$ as both $m$ and $t = \operatorname{Im}(s)$ go to infinity.