The structure of the zero set of a multivariate polynomial is a topic of wide interest, in view of its ubiquity in problems of analysis, algebra, partial differential equations, probability and geometry. The study of such sets, known in algebraic geometry literature as resolution of singularities, originated in the pioneering work of Jung, Abhyankar and Hironaka and has seen substantial recent advances, albeit in an algebraic setting. In this talk, I will discuss a few situations in analysis where the study of polynomial zero sets play a critical role, and discuss prior work in this analytical framework in two dimensions. Our main result (joint with Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for resolving singularities of a multivariate real-analytic function with a view to applying it to a class of problems in harmonic analysis.