Definitions: Flock, Star Flock, Herd Space, Herd


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 [back], 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 [back]. 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 [back], 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, Anot equal0, 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 mnot equal0, 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.


Return to Star Flocks II