Proposition 8.1.3: The herd space of a flock F contains a constant function, if and only if, F is a star flock.
Pf: If the herd space of F contains a constant function, fP, then we must have fP(t) = 0 for all t in GF(q), since all functions in the herd space are 0 at 0. By the definition of a herd space function, each plane must contain the point P of W = 0. Thus, F is a star flock.
Conversely, if F is a star flock, it is easy to see that the function of the herd space associated to a common point of all the planes of F is the zero function, i.e., a constant function.