2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114205Let $(f, g)$ be a pair of complex analytic functions on a singular analytic space $X$. We give ``the correct'' definition of the relative polar curve of $(f, g)$, and we give a very formal generalization of Lê's attaching result, which relates the relative polar curve to the relative cohomology of the Milnor fiber modulo a hyperplane slice. We also give the technical arguments which allow one to work with a derived category version of the discriminant and Cerf diagram of a pair of functions. From this, we derive a number of generalizations of results which are classically proved using the discriminant. In particular, we give applications to families of isolated ``critical points''.34 pages, the revision contains a new sectionAlgebraic Geometry32B15, 32C35, 32C18, 32B10text