Definitions: Flock, Critical Cone, Baselines, Star Flock, Herd Space, Herd, Linearized Polynomial

Proposition 11.1.1: If g is a pe-linearized polynomial, then g(t) - at is a permutation polynomial if and only if a is not of the form g(b)/b for any b in GF(q)\{0}.

Pf: Let E = GF(q) be the extension field of K = GF(pe) over which g is defined. In PG(3,q) construct the flock F(t,g(t),0). This is a star flock by Proposition 8.1.3 (of a possibly empty cone). The baselines of F pass through the point (0,0,1,0) and since g is an additive function, all secondary baselines coincide with primary baselines. The critical cone of F is thus the set of points other than (0,0,1,0) which are on the lines through (0,0,1,0) which are not baselines. In the herd space of F, the functions associated to the points of the critical cone are permutations (i.e., the herd of F). If g is a linear function, then F is a linear flock by Proposition 8.1.6 , and the condition is clearly necessary and sufficient. We may therefore assume that g is not linear, and F is a proper star flock. The baselines through (0,0,1,0) are the lines of W = 0 satisfying, tX + g(t)Y = 0, with t0. A point with coordinates (-a,1,0,0) is not on any baseline if and only if a is not of the form g(b)/b for any b in E\{0}. Thus, f(-a,1,0,0)(t) = -at + g(t) is a permutation if and only if a is not of the form g(b)/b for any b in E\{0}.