Newton-Puiseux roots of Jacobian determinants

Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A polar root is a Newton-Puiseux root, $x=\gamma (y)$, of the Jacobian $J=f_yg_x-f_xg_y$, but not a root of $f\cdot g$.

We define the tree-model, $T(f,g)$, for the pair, using the set of contact orders of the Newton-Puiseux roots of $f$ and $g$. Our main results (§2) describe how the $\gamma$’s climb, and leave, the tree (like vines). We also show by two examples (§5) that when the tree has what we call collinear points or bars, the way the $\gamma$’s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one function case where the tree $T(f)$ completely determines how the polar roots split away.

Our results yield a factorisation of the Jacobian determinant in $\mathbb {C} \{x,y\}$ (§6). As in the one-function case, the factors need not be invariants, nor irreducible. However, some factors do yield invariant truncations and intersection multiplicities (§7).