Definitions: Flock, Herd Space

Proposition 8.1.7: In the herd space of a flock F the functions associated with three collinear points of W = 0 are linearly dependent over GF(q). In particular, if the herd space contains two functions based at points P and Q which are scalar multiples of each other, then all the functions based at points on the line PQ are scalar multiples of the same function and F is a star flock.

Pf: Let P = (xP,yP,zP,0) and Q = (xQ,yQ,zQ,0) be two distinct points of W = 0, and let be the plane determined by V, P and Q. Let R be any point of the line PQ, other than P. The coordinates of R are given by (xQ + axP, yQ + ayP, zQ + azP, 0) for some a in GF(q). For each fixed t in GF(q), the plane of a flock F meets in a line mt which does not pass through V. mt meets the lines VP and VQ at the points (xP,yP,zP,-fP(t)) and (xQ,yQ,zQ,-fQ(t)), respectively. The points of mt, other than the intersection of mt and VP, have coordinates of the form (xQ + bxP, yQ + byP, zQ + bzP, -fP(t) - bfQ(t)), as b ranges through GF(q). The point of intersection of mt and the line VR is thus (xQ + axP, yQ + ayP, zQ + azP, -fP(t) - afQ(t)) = (xQ + axP, yQ + ayP, zQ + azP, -fR(t)). So, for each t we have fR(t) = fP(t) + afQ(t), and the functions fR, fP, fQ are linearly dependent for each choice of R. If fQ = kfP, then fR = (1 + ak)fP for some a in GF(q). If k = 0, then F is a star flock by Proposition 8.1.3 , otherwise let R' be the point on PQ with a = -1/k, then fR' is the zero function and F is a star flock.