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Star Flocks II

Star flocks have already been introduced [backbutton]. We first describe the structure of the herd space of a proper star flock in a manner similar to what was done for linear flocks (Proposition 8.1.4 [backbutton]).

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.

To examine proper star flocks more closely we will use the dual setting. As in the previous theorem, we shall assume that the flock is a flock of a non-empty cone and coordinatize so that the flock has the form F(t,g(t),0).[We shall convert to standard coordinates later, but for now this choice makes our discussion more easily comparable to the extant literature.] F is a proper star flock if and only if g is not linear (i.e., g(t) not equal kt, for some k in GF(q).) Since the planes of F all pass through the point (0,0,1,0), in the dual setting the points of DF all lie in the plane Z' = 0. This plane intersects the plane W' = 0 in the line m. We shall take m as the line at infinity of Z' = 0 and use affine coordinates in the affine plane obtained by removing m from Z' = 0. Thus, the point in this affine plane with coordinates (x,y) is the point (x,y,0,1) of PG(3,q). Hence, the q points of DF in this affine plane have coordinates (t,g(t)) for t in GF(q). Let M be the set of all "slopes" determined by pairs of these points, that is M = {(g(t)-g(s))/t-s | tnot equals, t,s in GF(q)}, and let N = |M|. Each slope in M corresponds to a point of m through which passes at least one line of Z' = 0 containing at least two points of DF. To satisfy the flock condition, no line of DG can pass through one of these points. Thus, if F is a proper star flock of a wide cone we must have N less than or equalq+1 -[q+2/2] = [q+1/2].

The problem of determining the number of slopes determined by q points in an affine plane has been well studied due to a connection with (line) blocking sets of Rédei type. This problem is almost completely settled and the relevant theorem (paraphrased in our terminology) is due to Blokhuis, Ball, Brouwer, Storme and Szönyi.

Theorem 11.1.2: If q = pn for some prime p, let e be the largest integer so that any line of the affine plane containing at least two points of DF contains a multiple of pe points of DF. Then we have one of the following:

  1. e = 0 and (q+3)/2 less than or equalNless than or equalq+1,
  2. p = 2 and e = 1, and (q+5)/3less than or equalNless than or equalq-1,
  3. pe > 2, then e | n and pn-e + 1less than or equalNless than or equal(q-1)/(pe -1),
  4. e = n and N = 1.
Moreover, if pe > 3 or (pe = 3 and N = q/3 + 1), then g is a pe-linearized polynomial. [reference] [BBBSS99]

A pe-linearized polynomial is one of the form:

In this theorem, all the bounds for N are sharp (there exist examples meeting the bounds) except for case (ii). In this case there is no known example with N < q/2 + 1, and it is conjectured that q/2 + 1 is the correct bound (known to be true for qless than or equal16).

The consequences of this theorem for star flocks of wide cones are as follows:

Theorem 11.1.3: If q = pn with p a prime > 3 then a proper star flock of any wide cone is given by F(t,g(t),0) if and only if g is a non-linear pe-linearized polynomial for some e|n with e < n.

Corollary 11.1.1: If q is a prime, then all star flocks of wide cones are linear.

Theorem 11.1.4: If q = pn, with p = 2 or 3 then a proper star flock of a wide cone is given by F(t,g(t),0) where g is a pe-linearized polynomial for some e|n with e < n or every line of Z' = 0 which contains at least two points of DF contains a multiple of p, but not a multiple of p2 points of DF and, in characteristic 3, N > q/3 + 1.

Note that there are no known examples of proper star flocks of a wide cone in characteristic 2 for which g is not a pe-linearized polynomial.

We examine two classes of examples which correspond to extreme values of N. Let E = GF( pn) with a subfield K = GF(pe), then the function g(t) = gives N = (q-1)/(pe -1) and for pe > 2 we have N < [q+1/2], and so, a star flock of a wide cone. These examples were known for quadratic cones in odd characteristic as Kantor (K1) flocks, or Kantor-Knuth flocks [forward][more reference:K1 flocks]. Under the same assumptions about q and e, the function g(t) = TrE/K(t), the relative trace function from E onto K, gives N = pn-e + 1, and again, if pe > 2, N < [q+1/2]. We call these proper star flocks of wide cones, Holder-Megeysi star flocks (L. Megeysi gave the example in the Rédei blocking set context and L. Holder, independently, gave the example in the conic blocking set context, see below).

If we consider the quadratic cone case, there are two fundamental theorems of J. Thas concerning star flocks.

Theorem 11.1.5: In even characteristic, all star flocks of quadratic cones are linear. [reference] [Th87]

Theorem 11.1.6: In odd characteristic, a star flock of a quadratic cone is either linear or a Kantor (K1) flock (only possible when the point corresponding to the constant function in the herd space is an exterior point of the conic which is the carrier of the cone). [reference] [Th87]

These theorems prove that the baselines of a proper star flock of a thick cone, which is not of Kantor (K1) type, form a (lineal) conic blocking set, that is, a set of lines of a plane which every conic in the plane must intersect. The conic blocking sets which arise in this way are proper, meaning that there is no conic totally contained in the set of lines, and consist of concurrent lines. Not all conic blocking sets satisfying these properties arise from star flocks. [more][more reference: conic blocking sets]

Finally, we consider the herds of these flocks. In the affine plane determined by removing the line Z = 0 from the plane W = 0, and using the affine coordinates (x,y) for the point with homogeneous coordinates (x,y,1,0), the lines through the point associated to the constant function (i.e., (0,0)) have the form y = mx or x = 0. By Theorem 11.1.1, we know that the functions associated to each of the points on these lines are just the different scalar multiples of a single function. With the flock F given by F(t,g(t),0), the function associated to the line x = 0 is g, while the function associated to y = mx is t + mg(t). Thus, the function f(t) = t is associated to the line y = 0, and we may write m-1t + g(t) for the function associated to the line y = mx, for mnot equal0. The primary baselines of F are the lines tx + g(t)y = 0, i.e., y = -(t/g(t))x if g(t)not equal0 and x = 0 otherwise. Now, assuming that g is a pe-linearized polynomial, all secondary baselines coincide with primary baselines, since g is an additive function. Thus, the critical cone of this flock (the points associated to the herd of the flock) consists of the points other than (0,0) on the lines y = mx, where m is not of the form -t/g(t) for t in GF(q). The functions of the herd are therefore the scalar multiples of g(t) - at, where a is not equal to g(b)/b for any b in GF(q). As these functions are permutations, we have:

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}.

Since g(t)-at is a linearized polynomial (i.e. p-linearized), and it is known that a linearized polynomial is a permutation polynomial if and only if its only root is 0, the above proposition is an obvious special case of this result.


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