Boundary local integrability of rational functions in two variables

Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on R² near $(0,0)$ of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to R²) at the origin. Building on work of Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a local Lᵖ(R²) space and we give a complete description of all numerators Q such that $Q/P$ is locally in a given Lᵖ space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to L¹ 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.