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 R 2 \mathbb {R}^{2} near ( 0 , 0 ) (0,0) of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to R 2 \mathbb {R}^{2} ) at the origin. Building on work of Bickel-Pascoe-Sola [Proc. Lond. Math. Soc. (3) 116 (2018), pp. 281–329; Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21 (2020), pp. 449–494], we give a necessary and sufficient test for membership in a local L p ( R 2 ) L^{\mathfrak {p}}(\mathbb {R}^{2}) space and we give a complete description of all numerators Q Q such that Q / P Q/P is locally in a given L p L^{\mathfrak {p}} space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to L 1 L^{1} on the two-torus. In addition, we give a new proof of a conjecture started by Bickel, Knese, Pascoe, and Sola [Ann. Polon. Math. 133 (2024), pp. 95–169] and completed by Kollár [Period. Math. Hungar. 88 (2024), pp. 1–7] characterizing the ideal of Q Q such that Q / P 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.