  Motivated by the recently observed phenomenon of topology trivialization of
potential energy landscapes (PELs) for several statistical mechanics models, we
perform a numerical study of the finite size $2$-spin spherical model using
both numerical polynomial homotopy continuation and a reformulation via
non-hermitian matrices. The continuation approach computes all of the complex
stationary points of this model while the matrix approach computes the real
stationary points. Using these methods, we compute the average number of
stationary points while changing the topology of the PEL as well as the
variance. Histograms of these stationary points are presented along with an
analysis regarding the complex stationary points. This work connects topology
trivialization to two different branches of mathematics: algebraic geometry and
catastrophe theory, which is fertile ground for further interdisciplinary
research.
