Singularities of multivariate Lagrangian aberration functions

With the advent of computer programs that compute aberration series to arbitrary orders, knowledge of the region of convergence of multivariate aberration series takes on greater practical importance. The convergence region is determined by the singularities of the aberration function, and this paper reports on an investigation of the cause, nature, and location of these singularities. For systems of homogeneous lenses and mirrors (whose shapes are contained in a certain extensive set), it is shown that there are four fundamental types of singularities in the aberration functions. The cause of the singularities is uncovered, and a set of conditions for locating all the singular points is derived. Perhaps most importantly, this knowledge of singularities gives an understanding of the unexpected breakdown of the convergence of aberration series (i.e., when there is no apparent physical cause) and is therefore invaluable in any attempt to remedy this situation.