Theorem 11.1.1: The functions of the herd space of a proper star flock consist of the zero function (at one point only), and all non-zero scalar multiples of q+1 other functions. The functions associated to the points on a line which passes through the point associated with the constant function are all scalar multiples of each other. If the herd of the flock is non-empty, we can coordinatize the plane so that the flock has the (non-standard) form F(t,g(t),0), in which case the q+1 functions are t and at + g(t) for a in GF(q). Moreover, if we remove the line Z = 0 from W = 0, and use affine coordinates in the resulting affine space, then the functions associated to the points on a line through the origin are all distinct.
Pf: Let F be a proper star flock. By Proposition 8.1.3 , the herd space of F contains a constant (hence, zero-) function, and since F is proper, it can only contain one by Proposition 8.1.4 . Suppose that this constant function is associated to the point (a,b,c,0) of W = 0. Let P and Q be two points on a line through (a,b,c,0), neither equal to this point. In the proof of Proposition 8.1.7 , it was shown that for each point R, other than P, on this line, fR = fP + AfQ, for some A in GF(q). With R = (a,b,c,0) we obtain, 0 = fP + AfQ. Since fP is not the zero function, A0, and fQ is a non-zero scalar multiple of fP.
Assuming that the herd of F is non-empty, there is at least one point of W = 0, whose associated function is a permutation. We can re-coordinatize W = 0 so that the function associated to the point (1,0,0,0) is a permutation, and the function associated to (0,0,1,0) is the zero function. We can then, as usual, re-parameterize the planes of the flock so that f(1,0,0,0)(t) = t. Thus, in terms of coordinate functions we will have F = F(t,g(t),0), where g(t) = f(0,1,0,0)(t). Now consider the points on the line Z = 0 in this plane. They have coordinates of the form (1,0,0,0) or (a,1,0,0) for each a in GF(q). f(a,1,0,0)(t) = at + g(t), and since one of these points is on each line through (0,0,1,0), by the previous paragraph, t together with at + g(t) are the q+1 functions whose scalar multiples are all of the herd functions appearing in the herd space of F.
Finally, we consider the affine plane determined by removing the line Z = 0 from the plane W = 0, and identify each point (x,y,1,0) with the point with affine coordinates (x,y). The points on the line y = mx (which passes through (0,0)) with m0, have associated functions xt + xmg(t) = xm(m-1t + g(t)). On y = 0, the functions are xt, and on x = 0 the functions are yg(t). Thus, in all cases the functions associated with the points on the lines through the origin are distinct multiples of the same function.