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.