Petters, AO; Witt, HJ, *Bounds on number of cusps due to point mass gravitational lenses*,
Journal of Mathematical Physics, vol. 37 no. 6
(1996),
pp. 2920-2933 [doi] .
**Abstract:**

*Generic caustics in gravitational lensing occur locally either as folds or cusps. This paper rigorously proves that the total number of cusps, Ncusps, due to g point masses on a single plane having non-normalized external shear γ>0 and continuous matter with constant density σc, is bounded as follows: 0≤Ncusps≤12g2. For vanishing shear γ=0 we obtain the result 0≤Ncusps≤12g(g-1). Consequences of these bounds for the global geometry of caustics are discussed. It is also shown that if γ≥0 and σc is sufficiently large, then all cusps can be eliminated, that is, Ncusps=0. The paper also includes equations for calculating all the bi-caustics (i.e., curves yielding the positions of cusps during a one-parameter evolution) of a single point-mass lens with continuous matter and shear. The methods of the paper are based on a new approach to point-mass gravitational lensing using complex quantities and the theory of resultants. © 1996 American Institute of Physics.*