This paper presents a criterion for uniqueness of a critical point in $H_{2,\RR}$ rational approximation of type $(m,n)$, with $m\geq n-1$. This criterion is differential topologic in nature, and turns out to be connected with corona equations and classical interpolation theory. We illustrate its use on three examples, namely best approximation of fixed type on small circles, a de Montessus de Ballore type theorem, and diagonal approximation to the exponential function of large degree.