One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) s
olutions but they each exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor $phi^4$ model and atomic clusters.
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 pol
ynomial 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.
