Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics

Abstract:

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