On Puiseux roots of Jacobians

Abstract

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.