  Motivated by studying boundary singularities of rational functions in two
variables that are analytic on a domain, we investigate local integrability on
$\mathbb{R}^2$ near $(0,0)$ of rational functions with denominator
non-vanishing in the bi-upper half-plane but with an isolated zero (with
respect to $\mathbb{R}^2$) at the origin. Building on work of
Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a
local $L^{p}(\mathbb{R}^2)$ space and we give a complete description of all
numerators $Q$ such that $Q/P$ is locally in a given $L^{p}$ space. As
applications, we prove that every bounded rational function on the bidisk has
partial derivatives belonging to $L^1$ on the two-torus. In addition, we give a
new proof of a conjecture, started in Bickel-Knese-Pascoe-Sola and completed by
Koll\'ar, characterizing the ideal of $Q$ such that $Q/P$ is locally bounded. A
larger takeaway from this work is that a local model for stable polynomials we
employ is a flexible tool and may be of use for other local questions about
stable polynomials.
