  We study the critical points of the likelihood function over the Fermat
hypersurface. This problem is related to one of the main problems in
statistical optimization: maximum likelihood estimation. The number of critical
points over a projective variety is a topological invariant of the variety and
is called maximum likelihood degree. We provide closed formulas for the maximum
likelihood degree of any Fermat curve in the projective plane and of Fermat
hypersurfaces of degree 2 in any projective space. Algorithmic methods to
compute the ML degree of a generic Fermat hypersurface are developed throughout
the paper. Such algorithms heavily exploit the symmetries of the varieties we
are considering. A computational comparison of the different methods and a list
of the maximum likelihood degrees of several Fermat hypersurfaces are available
in the last section.
