  The stationary points (SPs) of the potential energy landscapes (PELs) of
multivariate random potentials (RPs) have found many applications in many areas
of Physics, Chemistry and Mathematical Biology. However, there are few reliable
methods available which can find all the SPs accurately. Hence, one has to rely
on indirect methods such as Random Matrix theory. With a combination of the
numerical polynomial homotopy continuation method and a certification method,
we obtain all the certified SPs of the most general polynomial RP for each
sample chosen from the Gaussian distribution with mean 0 and variance 1. While
obtaining many novel results for the finite size case of the RP, we also
discuss the implications of our results on mathematics of random systems and
string theory landscapes.
