Definitions: Flock, Star Flock, Coordinate Functions


Theorem 7.1: A flock is a star flock if and only if the coordinate functions are linearly dependent over GF(q).

Pf: Let F = F(f,g,h) be a star flock. If P is a point in common with all the planes of F, then P must lie in W = 0, and so, P has coordinates (a,b,c,0) with not all of a, b and c being 0. Since this point lies in every plane of F, we have that:

af(t) + bg(t) + ch(t) = 0, for all t in GF(q).
Thus, the coordinate functions are linearly dependent over GF(q).

Conversely, if the coordinate functions are linearly dependent over GF(q), then there exist elements A, B, and C in GF(q), not all 0, such that:

Af(t) + Bg(t) +Ch(t) = 0, for all t in GF(q).
The point (A,B,C,0) thus satisfies the equations f(t)X + g(t)Y + h(t)Z + W = 0 for all t, and so this point is incident with each plane of F, i.e., F is a star flock.


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