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 $z_1+\cdots +z_{\mu }+1=0$.