Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics

Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We give a formula of the Maximum likelihood degree in terms of the constants $\beta_{\mu, \nu}$, which is defined to be the number of complex solutions to the system of equations $z_1^\nu=z_2^\nu=\cdots=z_\mu^\nu=1$ and $z_1+\cdots +z_\mu+1=0$.