The number of directions determined by less than q points

Abstract

In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187–196, 1999) on the number of directions determined by q affine points.

1 Introduction

In this article, p is a prime, and q=p ^h, where h≥1. GF(q) denotes the finite field with q elements, and $\mathbb {F}$ can denote an arbitrary field (or maybe a Euclidean ring). PG(d,q) denotes the projective geometry of dimension d over the finite field GF(q). AG(d,q) denotes the affine geometry of dimension d over GF(q) that corresponds to the coordinate space GF(q)^d of rank d over GF(q).

For the affine and projective planes AG(2,q)⊂PG(2,q), we imagine the line ℓ _∞=PG(2,q)∖AG(2,q) at infinity as the set ℓ _∞≅GF(q)∪{∞}. So the nonvertical directions are field elements (numbers), and the vertical direction is ∞.

The original problem of direction sets was the following. Let f:GF(q)→GF(q) be a function, and let $\mathcal {U}=\{(x,f(x))\;|\;x\in \mathsf {GF}(q)\}\subseteq \mathsf {AG}(2,q)$ be the graph of the function f. The question is how many directions can be determined by the graph of f?

Definition

(Direction set)

If $\mathcal {U}$ is an arbitrary set of points in the affine plane AG(2,q), then we say that the set

$$\mathcal {D}= \biggl\{\frac{b-d}{a-c}\biggm{|}(a,b),(c,d)\in \mathcal {U},(a,b)\neq(c,d) \biggr\}$$

is the set of directions determined by $\mathcal {U}$. We define $\frac{a}{0}$ as ∞ if a≠0; thus $\mathcal {D}\subseteq \mathsf {GF}(q)\cup\{\infty\}$. If $\mathcal {U}$ is the graph of a function, then it simply means that $|\mathcal {U}|=q$ and $\infty\notin \mathcal {D}$.

Simeon Ball [1] proved a stronger version of the structure theorem of Aart Blokhuis, Simeon Ball, Andries Brouwer, Leo Storme and Tamás Szőnyi [3]. To recall their result, we need some definitions.

Definition

Let $\mathcal {U}$ be a set of points of AG(2,q). If y∈ℓ _∞ is an arbitrary direction, then by s(y) we denote the greatest power of p such that each line ℓ of direction y meets $\mathcal {U}$ in zero modulo s(y) points. In other words,

$$s(y)=\gcd\bigl( \bigl\{|\ell\cap \mathcal {U}|\mid\ell\cap\ell_\infty =\{y\}\bigr\}\cup\bigl\{p^h \bigr\} \bigr).$$

Let s be the greatest power of p such that each line ℓ of direction in $\mathcal {D}$ meets $\mathcal {U}$ in zero modulo s points. In other words,

$$s=\gcd_{y\in \mathcal {D}}s(y)=\min_{y\in \mathcal {D}}s(y).$$

Note that s(y) and thus also s may be equal to 1. Note that s(y)=1 for each nondetermined direction $y\notin \mathcal {D}$.

Remark 1

Suppose that s≥p. Then, for each line ℓ⊂PG(2,q):

$$\begin{array}{@{}l@{\quad}l}\mbox{either} & (\mathcal {U}\cup \mathcal {D})\cap\ell=\emptyset;\\[5pt]\mbox{or} & \bigl \vert (\mathcal {U}\cup \mathcal {D})\cap\ell\bigr \vert \equiv1\pmod{s}.\end{array}$$

Moreover, $|\mathcal {U}|\equiv0\pmod{s}$.

(If s=1, then 0≡1(mods), so in this case the remarks above would be meaningless.)

Proof

Fix a direction $y\in \mathcal {D}$. Each affine line with slope y meets $\mathcal {U}$ in zero modulo s points, so $|\mathcal {U}|\equiv0\pmod{s}$.

An affine line L⊂AG(2,q) with slope $y\in \mathcal {D}$ meets $\mathcal {U}$ in 0(mods) points, so the projective line ℓ=L∪{y} meets $\mathcal {U}\cup \mathcal {D}$ in 1(mods) points.

An affine line L⊂AG(2,q) with slope $y\notin \mathcal {D}$ meets $\mathcal {U}$ in at most one point, so the projective line ℓ=L∪{y} meets $\mathcal {U}\cup \mathcal {D}$ in either zero or one point.

Let $P\in \mathcal {U}$, and let L⊂AG(2,q) an affine line with slope $y\in \mathcal {D}$ such that P∈L. Then the projective line ℓ=L∪{y} meets $\mathcal {U}$ in 0(mods) points, and thus, ℓ meets $\mathcal {D}\cup \mathcal {U}\setminus\{P\}$ also in 0(mods) points. Thus, considering all the lines through P (with slope in $\mathcal {D}$), we get $|\mathcal {U}\cup \mathcal {D}|\equiv1\pmod{s}$. Since $\mathcal {U}$ has 0(mods) points, $|\mathcal {D}|\equiv1\pmod{s}$. So we get that $\mathcal {U}\cup \mathcal {D}$ meets also the ideal line in 1(mods) points. □

Remark 2

(Blocking set of Rédei type)

If $|\mathcal {U}|=q$, then each of the q affine lines with slope $y\notin \mathcal {D}$ meets $\mathcal {U}$ in exactly one point, so $\mathcal {B}=\mathcal {U}\cup \mathcal {D}$ is a blocking set meeting each projective line in 1(mods) points. Moreover, if $\infty\notin \mathcal {D}$, then $\mathcal {U}$ is the graph of a function, and in this case the blocking set $\mathcal {B}$ above is said to be of Rédei type.

Theorem 3

(Blokhuis, Ball, Brouwer, Storme, and Szőnyi [3]; Ball [1], Theorem 1.1)

Let $|\mathcal {U}|=q$ and $\infty\notin \mathcal {D}$. Using the notation s defined above, one of the following holds:

$$\everymath{\displaystyle}\begin{array}{@{}l@{\quad}l@{\quad}l@{\quad}l}\mathrm{either} & s=1 & \mbox{\textit{and}} & \frac{q+3}{2}\le \vert \mathcal {D}\vert \le q;\\[8pt]\mathrm{or} & \mathsf {GF}(s) \mbox{ is a subfield of }\mathsf {GF}(q) &\mbox{\textit{and}} & \frac{q}{s}+1\le \vert \mathcal {D}\vert \le \frac{q-1}{s-1};\\[10pt]\mathrm{or} & s=q & \mbox{\textit{and}} & |\mathcal {D}|=1.\end{array}$$

Moreover, if s>2, then $\mathcal {U}$ is a GF(s)-linear affine set (of rank log _s q).

Definition

(Affine linear set)

A GF(s)-linear affine set is the GF(s)-linear span of some vectors in $\mathsf {AG}(n,q)\cong \mathsf {GF}(s^{\log_{s} q})^{n}\cong \mathsf {GF}(s)^{n\log_{s} q}$ (or possibly a translate of such a span). The rank of the affine linear set is the rank of this span over GF(s).

What about the directions determined by an affine set $\mathcal {U}\subseteq \mathsf {AG}(2,q)$ of cardinality not q? Using the pigeon-hole principle, one can easily prove that if $|\mathcal {U}|>q$, then it determines all the q+1 directions. So we can restrict our research to affine sets of less than q points.

Examining the case q=p prime, Tamás Szőnyi [7] and later (independently) also Aart Blokhuis [2] have proved the following result.

Theorem 4

(Szőnyi [7], Theorem 5.2; Blokhuis [2])

Let q=p prime and suppose that $1<|\mathcal {U}|\le p$. Also suppose that $\infty\notin \mathcal {D}$. Then

$$\everymath{\displaystyle}\begin{array}{@{}l@{\quad}l}\mathrm{either} & \frac{|\mathcal {U}|+3}{2}\le \vert \mathcal {D}\vert \le p; \\[10pt]\mathrm{or} & \mathcal {U}\mbox{ \textit{is collinear}}\quad\mbox{\textit{and}}\quad \vert \mathcal {D}\vert =1.\end{array}$$

Moreover, these bounds are sharp.

In this article we try to generalize this result to the q=p ^h prime power case by proving an analogue of Theorem 3 for the case $|\mathcal {U}|\le q$. Before we examine the number of directions determined by less than q affine points in the plane, we ascend from the plane in the next section and examine the connection between linear sets and direction sets in arbitrary dimensions. The further sections will return to the plane.

2 Linear sets as direction sets

The affine space AG(n,q) and its ideal hyperplane Π _∞≅PG(n−1,q) of directions together constitute a projective space PG(n,q). We say that the point P∈Π _∞ is a direction determined by the affine set $\mathcal {U}\subset \mathsf {AG}(n,q)$ if there exists at least one line through P that meets $\mathcal {U}$ in at least two points.

Definition

(Projective linear set)

Suppose that GF(s) is a subfield of GF(q). A projective GF(s)-linear set $\mathcal {B}$ of rank d+1 is a projected image of the canonical subgeometry PG(d,s)⊂PG(d,q) from a center disjoint to this subgeometry. The projection can yield multiple points.

Proposition 5

Suppose that $\mathcal {U}$ is an affine GF(s)-linear set of rank d+1 in AG(n,q) such that AG(n,q) is the smallest dimensional affine subspace that contains $\mathcal {U}$. Let $\mathcal {D}$ denote the set of directions determined by $\mathcal {U}$. The set $\mathcal {U}\cup \mathcal {D}$ is a projective GF(s)-linear set of rank d+1 in PG(n,q), and all the multiple points are in $\mathcal {D}$.

Proof

Without loss of generality, we can suppose that $\mathcal {U}$ contains the origin and suppose that $\mathcal {U}$ is the set of GF(s)-linear combinations of the vectors a ₀,a ₁,…,a _d . We can coordinatize AG(n,q) so that a _d−n+1,…,a _d is the standard basis of GF(q)ⁿ≅AG(n,q).

Embed GF(q)ⁿ≅AG(n,q) into GF(q)^d+1≅AG(d+1,q) so that z ₀,z ₁,…,z _d−n ,a _d−n+1,…,a _d is the standard basis. Let π denote the projection of AG(d+1,q) onto AG(n,q) such that π(z _i )=a _i for each i=0,…,d−n and π(a _j )=a _j for each j>d−n. Then $\mathcal {U}$ is the image of the canonical subgeometry AG(d+1,s) by π.

Extend π to the ideal hyperplane. The extended $\bar{\pi}$ is a collineation, and so the image of a determined direction is a determined direction. Vice versa, let A and B two arbitrary distinct points in $\ell\cap \mathcal {U}$, and let P be the direction determined by π ⁻¹(A) and π ⁻¹(B), then the direction of ℓ is $\bar{\pi}(P)$. □

Corollary 6

If $\mathcal {D}$ is the set of directions determined by an affine GF(s)-linear set, then $\mathcal {D}$ is a projective GF(s)-linear set.

Remark 7

Olga Polverino [4, Proposition 2.2] proved that if $\mathcal {D}$ is a projective GF(s)-linear set, then $|\mathcal {D}|\equiv1\pmod{s}$.

The proposition above says that the set of directions determined by an affine linear set is a projective linear set. The converse of this proposition is also true; each projective linear set is a direction set:

Theorem 8

Embed PG(n,q) into PG(n+1,q) as the ideal hyperplane and let AG(n+1,q)=PG(n+1,q)∖PG(n,q) denote the affine part. For each projective GF(s)-linear set $\mathcal {D}$ of rank d+1 in PG(n,q), there exists an affine GF(q)-linear set $\mathcal {U}$ of rank d+1 in AG(n+1,q) such that the set of directions determined by $\mathcal {U}$ is $\mathcal {D}$.

Proof

$\mathcal {D}\subset \mathsf {PG}(n,q)$ is the image of the canonical subgeometry PG(d,s)⊂PG(d,q) by the projection π:PG(d,q)→PG(n,q), where the center C of π is disjoint to this subgeometry. Embed PG(d,q) into PG(d+1,q) as the ideal hyperplane and extend π to $\bar{\pi}:\mathsf {PG}(d\!+\!1,q)\rightarrow \mathsf {PG}(n\!+\!1,q)$ so that its center remains C. That is, the center is in the ideal hyperplane. Consider the canonical subgeometry PG(d+1,s)⊂PG(d+1,q) and its image by $\bar{\pi}$:

The “ideal part” of this canonical subgeometry PG(d,s) is the original canonical subgeometry PG(d,s) of PG(d,q), and the projection $\bar{\pi}$ project this onto $\mathcal {D}$. Since the center is totally contained in the ideal hyperplane, $\bar{\pi}$ maps the affine part of the canonical subgeometry PG(d+1,s) one-to-one.

The directions determined by the affine part of PG(d+1,s) are the points of PG(d,s) in the ideal hyperplane of AG(d+1,q). Since the extended $\bar{\pi}$ preserves collinearity, the set of directions determined by the projected image of the affine part is $\mathcal {D}$. □

3 The Rédei polynomial of less than q points

Let $\mathcal {U}$ be a set of less than q affine points in AG(2,q), and let $\mathcal {D}$ denote the set of directions determined by $\mathcal {U}$. Let $n=|\mathcal {U}|$, and let R(X,Y) be the inhomogeneous affine Rédei polynomial of the affine set $\mathcal {U}$, that is,

$$R(X,Y)=\prod_{(a,b)\in \mathcal {U}}(X-aY+b)=X^n+\sum _{i=0}^{n-1}\sigma_{n-i}(Y)X^{i},$$

where the abbreviation σ _k (Y) means the kth elementary symmetric polynomial of the set $\{b-aY\;|\;(a,b)\in \mathcal {U}\}$ of linear polynomials.

Proposition 9

If $y\in \mathcal {D}$, then R(X,y)∈GF(q)[X ^s(y)]∖GF(q)[X ^p⋅s(y)].

If $y\notin \mathcal {D}$, then R(X,y)∣X ^q−X.

Proof

Let $y\in \mathcal {D}$. Then the equation R(X,y)=0 has a root x with multiplicity m if there is a line with slope y meeting $\mathcal {U}$ in exactly m points. The value of x determines this line. So each x is either not a root of R(X,y) or a root with multiplicity a multiple of s(y), and p⋅s(y) does not have this property. Since R is totally reducible, it is the product of its root factors.

If $y\notin \mathcal {D}$, then a line with direction y cannot meet $\mathcal {U}$ in more than one point, so x cannot be a multiple root of R(X,y). □

Notation

Let $\mathbb {F}$ be the polynomial ring GF(q)[Y] and consider R(X,Y) as the element of the univariate polynomial ring $\mathbb {F}[X]$. Divide X ^q−X by R(X,Y) as a univariate polynomial over $\mathbb {F}$; let Q denote the quotient, and let H+X be the negative of the remainder:

So

$$R(X,Y)Q(X,Y)=X^q+H(X,Y)=X^q+\sum _{i=0}^{q-1}h_{q-i}(Y)X^i,$$

where deg _X H<deg _X R. Let σ ^∗ denote the coefficients of Q,

$$Q(X,Y)=X^{q-n}+\sum_{i=0}^{q-n-1}\sigma^*_{q-n-i}(Y)X^{i},$$

and so

$$h_j(Y)=\sum_{i=0}^{j}\sigma_i(Y)\sigma^*_{j-i}(Y).$$

We know that degh _i ≤i, degσ _i ≤i, and $\deg\sigma^{*}_{i}\le i$. Note that the σ ^∗(Y) polynomials are not necessarily elementary symmetric polynomials of linear polynomials and if $y\in \mathcal {D}$, then Q(X,y) is not necessarily totally reducible.

Remark 10

Since deg _X H<deg _X R, we have h _i =0 for 1≤i≤q−n. By definition, $\sigma_{0}=\sigma^{*}_{0}=1$. The equation h ₁=0 implies $\sigma^{*}_{1}=-\sigma_{1}$; this fact and the equation h ₂=0 imply $\sigma^{*}_{2}=-\sigma_{2}+\sigma_{1}^{2}$ and so on, and the q−n equations h _i =0 uniquely define all the coefficients $\sigma^{*}_{i}$.

Proposition 11

If $y\in \mathcal {D}$, then Q(X,y),H(X,y)∈GF(q)[X ^s(y)], and if degR≤degQ, then Q(X,y)∈GF(q)[X ^s(y)]∖GF(q)[X ^p⋅s(y)].

If $y\notin \mathcal {D}$, then R(X,y)Q(X,y)=X ^q+H(X,y)=X ^q−X. In this case Q(X,y) is also a totally reducible polynomial.

Proof

If $y\in \mathcal {D}$, then

$$R(X,y)=X^n+\sum_{i=0}^{n-1}\sigma_{n-i}(y)X^{i} \in \mathsf {GF}(q)\bigl[X^{s(y)}\bigr]\setminus \mathsf {GF}(q)\bigl[X^{p\cdot s(y)}\bigr].$$

So s(y)∣n and σ _i (y)≠0 implies s(y)∣n−i and hence s(y)∣i. The defining equation of $\sigma^{*}_{i}$ contains the sum of products of some σ _j , where the sum of indices (counted with multiplicities) is i. Since σ _j (y)≠0 only if s(y)∣j, also $\sigma^{*}_{i}(y)\neq0$ only if s(y)∣i.

If degR≤degQ, then we can consider R as $(X^{q}-X)\operatorname {div}Q$, and the reminder is the same H.

Since both R(X,y) and Q(X,y) are in GF(q)[X ^s(y)], H(X,y)∈GF(q)[X ^s(y)].

If $y\notin \mathcal {D}$, then R(X,y)∣(X ^q−X) in GF(q)[X], so Q(X,y) is also totally reducible. □

Remark 12

Note that H(X,y) can be an element of GF(q)[X ^p⋅s(y)]. If H(X,y)≡a is a constant polynomial, then R(X,y)Q(X,y)=X ^q+a=X ^q+a ^q=(X+a)^q. This means that R(X,y)=(X+a)ⁿ, and thus, there exists exactly one line (corresponding to X=−a) of direction y that contains $\mathcal {U}$, and so $\mathcal {D}=\{y\}$.

Definition

If $|\mathcal {D}|\ge2$ (i.e., H(X,y) is not a constant polynomial), then for each $y\in \mathcal {D}$, let t(y) denote the maximal power of p such that H(X,y)=f _y (X)^t(y) for some f _y (X)∉GF(q)[X ^p]:

$$H(X,y)\in \mathsf {GF}(q)\bigl[X^{t(y)}\bigr]\setminus \mathsf {GF}(q)\bigl[X^{t(y)p}\bigr].$$

In this case, t(y)<q since t(y)≤deg _X H<q. Let t be the greatest common divisor of the numbers t(y), that is,

$$t=\gcd_{y\in \mathcal {D}}t(y)=\min_{y\in \mathcal {D}}t(y).$$

If H(X,y)≡a (i.e., $\mathcal {D}=\{y\}$), then we define t=t(y)=q.

Remark 13

If there exists at least one determined direction $y\in \mathcal {D}$ such that H(X,y) is not constant, then t<q. From Proposition 11 we have s(y)≤t(y) for all $y\in \mathcal {D}$, so s≤t.

Proposition 14

Using the notation above,

$$R(X,Y)Q(X,Y)=X^q+H(X,Y)\in \operatorname {Span}_\mathbb {F}\bigl\langle 1,X,X^{t},X^{2t},X^{3t},\dots,X^{q}\bigr\rangle.$$

Proof

If $|\mathcal {D}|=\{y\}$, then H(X,y)≡a and H(X,z)=−X for z≠y.

Suppose that $|\mathcal {D}|\ge2$. If $y\notin \mathcal {D}$, then X ^q+H(X,y)=X ^q−X, and if $y\in \mathcal {D}$, then X ^q+H(X,y)∈GF(q)[X ^t(y)]∖GF(q)[X ^t(y)p].

Thus, in both cases, if i≠1 and i∤t, then h _q−i (Y) has q roots, and its degree is less than q. □

4 Bounds on the number of directions

Although, in the original problem, the vertical direction ∞ was not determined, from now on, without loss of generality, we suppose that ∞ is a determined direction (if not, we apply an affine collineation). We continue to suppose that there is at least one nondetermined direction.

Lemma 15

If $\infty\in \mathcal {D}\subsetneqq\ell_{\infty}$, then $|\mathcal {D}|\ge\deg_{X} H(X,Y)+1$.

Proof

If $y\notin \mathcal {D}$, then R(X,y)∣X ^q−X; therefore, H(X,y)=−X, and thus ∀i≠q−1: h _i (y)=0.

If $y\in \mathcal {D}$, then R(X,y)∤X ^q−X; hence, ∃i≠q−1: h _i (y)≠0, and thus $h_{i}\not\equiv0$. Let i be the smallest index such that $h_{i}\not\equiv0$, and so i=q−deg _X H. Since $h_{i}\not\equiv0$ has at least $(q+1)-|\mathcal {D}|=q-(|\mathcal {D}|-1)$ roots, $\deg_{Y}h_{i}\ge q-|\mathcal {D}|+1$.

$$\mbox{Now } q\ge\deg X^{q-i}h_i(Y)=q-i+\deg_Yh_i\ge2q-|\mathcal {D}|+1-i. \hfill $$

Hence $|\mathcal {D}|\ge q+1-i=\deg_{X} H+1$. □

Lemma 16

Let κ(y) denote the number of the roots of X ^q+H(X,y) in GF(q), counted with multiplicity. If X ^q+H(X,y)≠X ^q−X and if H(X,y) is not a constant polynomial, then

$$\frac{\kappa(y)-1}{t(y)+1}+1= \frac{\kappa(y)+t(y)}{t(y)+1}\le t(y)\cdot\deg f_y(X)=\deg_X H \le\deg H.$$

Proof

Fix $y\in \mathcal {D}$ and utilize that X ^q+H(X,y)∈GF(q)[X ^t(y)]; thus,

$$\bigl(X^{q/t(y)}+f_y(X) \bigr)^{t(y)}=X^q+H(X,y)=\bigl(a(X)\cdot b(X)\cdot c(X) \bigr)^{t(y)},$$

where the totally reducible a(X) contains all the roots (in GF(q)) without multiplicity, the totally reducible b(X) contains the further roots (in GF(q)), and c(X) has no root in GF(q). (Note that t(y)<q, so X ^q/t(y)∈GF(q)[X ^p].)

So, deg(a(X)b(X))≤t(y)⋅degf _y +degf _y −1=(t(y)+1)⋅degf _y −1, since ∂ _X f _y (X)≠0 and f _y (X)^t(y)=H(X,y)≠−X. We get

$$\frac{\kappa(y)+t(y)}{t(y)+1}= t(y)\frac{\deg (a(X)b(X))+1}{t(y)+1}\le t(y)\cdot\deg f_y(X)$$

using κ(y)=t(y)⋅deg(a(X)b(X)). □

Using these lemmas above, we can prove a theorem similar to Theorem 3 but it is weaker in our case.

Theorem 17

Let $\mathcal {U}\subset \mathsf {AG}(2,q)$ be an arbitrary set of points, and let $\mathcal {D}$ denote the directions determined by $\mathcal {U}$. We use the notation s and t defined above geometrically and algebraically, respectively. Suppose that $\infty\in \mathcal {D}$. One of the following holds:

$$\everymath{\displaystyle}\begin{array}{@{}l@{\quad}l@{\quad}l@{\quad}l}\mathrm{either} & 1=s\le t<q & \mbox{\textit{and}} & \frac{|\mathcal {U}|-1}{t+1}+2\le \vert \mathcal {D}\vert \le q+1;\\[10pt]\mathrm{or} & 1<s\le t<q & \mbox{\textit{and}}& \frac{|\mathcal {U}|-1}{t+1}+2\le \vert \mathcal {D}\vert \le\frac{|\mathcal {U}|-1}{s-1};\\[10pt]\mathrm{or} & 1\le s\le t=q& \mbox{\textit{and}} & \mathcal {D}=\{\infty\}.\end{array}$$

Proof

The third case is trivial (t=q means $|\mathcal {D}|=1$, by the definition of t).

Let P be a point of $\mathcal {U}$ and consider the lines connecting P and the ideal points of $\mathcal {D}$. Since each such line meets $\mathcal {U}$ and has a direction determined by $\mathcal {U}$, it is incident with $\mathcal {U}$ in a multiple of s points. If s>1, then counting the points of $\mathcal {U}$ on these lines, we get the upper bound.

If t<q, then we can choose a direction $y\in \mathcal {D}$ such that the conditions of Lemma 16 hold. Using Lemma 15 and Lemma 16, we get

$$|\mathcal {D}|\ge\deg_X H(X,Y)+1\ge\frac{\kappa(y)-1}{t(y)+1}+1+1.$$

The number of roots of R(X,y)Q(X,y) is at least the number of roots of R(X,y), which equals $|\mathcal {U}|$. Therefore, $\kappa(y)\ge|\mathcal {U}|$, and thus,

$$\frac{\kappa(y)-1}{t(y)+1}\ge\frac{|\mathcal {U}|-1}{t+1}.$$

 □

An affine collineation converts Szőnyi’s and Blokhuis’ Theorem 4 to the special case of our Theorem 17, since t is equal to either 1 or p in the case q=p prime.

In the case q>p, the main problem of Theorem 17 is that the definition of t is nongeometrical. Unfortunately, t=s does not hold in general. For example, let $\mathcal {U}$ be a GF(p)-linear set minus one point. In this case, s=1, but t=p. In the rest of this article, we try to describe this problem.

5 Maximal affine sets

One can easily show that a proper subset of the affine set $\mathcal {U}$ can determine the same directions. (For example, let $\mathcal {U}$ be an affine subplane over the subfield GF(s). Arbitrary s+1 points of $\mathcal {U}$ determine the same directions.)

Definition

(Maximal affine set)

We say that $\mathcal {U}\subseteq \mathsf {AG}(2,q)$ is a maximal affine set that determines the set $\mathcal {D}\subseteq\ell_{\infty}\cong \mathsf {PG}(1,q)$ of directions if each affine set that contains $\mathcal {U}$ as a proper subset determines more than $|\mathcal {D}|$ directions.

Tamás Szőnyi proved a “completing theorem” (stability result) in [6], which was slightly generalized in [5] as follows.

Theorem 18

(Szőnyi [6]; Sziklai [5], Theorem 3.1)

Let $\mathcal {D}$ denote the set of directions determined by the affine set $\mathcal {U}\subset \mathsf {AG}(2,q)$ containing q−ε points, where $\varepsilon<\alpha\sqrt{q}$ and $|\mathcal {D}| < (q+1)(1-\alpha)$, 1/2<α<1. Then $\mathcal {U}$ can be extended to a set $\mathcal {U}'$ with $|\mathcal {U}'|=q$ such that $\mathcal {U}'$ determines the same directions.

Szőnyi’s stability theorem above also stimulates us to restrict ourselves to examine the maximal affine sets only. (An affine set of q points that does not determine all directions is automatically maximal.)

If we examine polynomials in one variable instead of Rédei polynomials, we can get similar “stability” results. Such polynomials occur when we examine R(X,y), Q(X,y), and H(X,y) or R, Q, and H over GF(q)(Y). The second author conjectured that if “almost all” roots of a polynomial g∈GF(q)[X] have muliplicity a power of p, then the quotient $X^{q}\operatorname {div}g$ extends g to a polynomial in GF(q)[X ^p]. We can prove more.

Notation

Let $p=\operatorname {char}\mathbb {F}\neq0$ be the characteristic of the arbitrary field $\mathbb {F}$. Let s=p ^e and q=p ^h two arbitrary powers of p such that e≤h (i.e., s∣q, but q is not necessarily a power of s).

Theorem 19

Let $g,f\in \mathbb {F}[X]$ be polynomials such that $g\cdot f\in \mathbb {F}[X^{s}]$. If 0≤degf≤s−1, then $X^{q}\operatorname {div}g$ extends g to a polynomial in $\mathbb {F}[X^{s}]$.

Proof

We know that deg(gf)=ks (k∈ℕ). Let $r=X^{q}\ \operatorname{mod}(fg)$ denote the remainder, that is, X ^q=(gf)h+r where degr≤deg(fg)−1.

Now we show that $h\in \mathbb {F}[X^{s}]$. Suppose to the contrary that $h\notin \mathbb {F}[X^{s}]$, i.e., the polynomial h has at least one monomial $\bar{a}\notin \mathbb {F}[X^{s}]$. Let $\bar{a}$ denote such a monomial of maximal degree. Let $\bar{b}$ denote the leading term of fg. Since $gf\in \mathbb {F}[X^{s}]$, also $\bar{b}\in \mathbb {F}[X^{s}]$, and since $\bar{b}$ is the leading term, $\deg\bar{b}=\deg(fg)$. The product $\bar{a}\bar{b}$ is a monomial of the polynomial (fg)h, and since $\deg\bar{b}>\deg r$, $\bar{a}\bar{b}$ is also a monomial of the polynomial (f⋅g⋅h+r). The monomial $\bar{a}\bar{b}$ is not in $\mathbb {F}[X^{s}]$ because $\bar{a}\notin \mathbb {F}[X^{s}]$ and $\bar{b}\in \mathbb {F}[X^{s}]$. But $f\cdot g\cdot h+r=X^{q}\in \mathbb {F}[X^{s}]$, which is a contradiction.

Hence $r\in \mathbb {F}[X^{s}]$, and so s∣degr; thus, if the closed interval [degg,deg(gf)−1] does not contain any integer that is a multiple of s, then degr is less than degg.

If we know that degr<degg, then from the equation X ^q=(gf)h+r we get X ^q=g(fh)+r; hence $r=X^{q}\mod g$ and $X^{q} \operatorname {div}g = f\cdot h$, where f is a polynomial such that $fg\in \mathbb {F}[X^{s}]$ and also $h\in \mathbb {F}[X^{s}]$.

So it is enough to show that the closed interval [degg,deg(gf)−1] does not contain any integer that is a multiple of s. Using deg(fg)=ks, we have that the closed interval [degg,deg(gf)−1]=[ks−degf,ks−1], and if 0≤degf≤s−1, then it does not contain any integer which is a multiple of s. □

This theorem above suggests that if the product R(X,y)Q(X,y) is an element of GF(q)[Y][X ^p⋅s(y)] while R(X,y)∈GF(q)[Y][X ^s(y)]∖GF(q)[Y][X ^p⋅s(y)], then a “completing result” might be in the background. If $\mathcal {U}$ is a maximal affine set, then it cannot be completed, so we conjecture the following.

Conjecture 20

If $\mathcal {U}$ is a maximal affine set that determines the set $\mathcal {D}$ of directions, then t(y)=s(y) for all $y\in \mathcal {D}$ where t(y)>2.

Note that there can be maximal affine sets which are not linear.

Example

(Nonlinear maximal affine set)

Let $\mathcal {U}\subset \mathsf {AG}(2,q)$ be a set, $|\mathcal {U}|=q$, s=1, $q\ge|\mathcal {D}|\ge\frac{q+3}{2}$. In this case, $\mathcal {U}$ cannot be linear because then s would be at least p. But $\mathcal {U}$ must be maximal since q+1 points in AG(2,q) would determine all directions. Embed AG(2,q) into AG(2,q ^m) as a subgeometry. Then $\mathcal {U}\subset \mathsf {AG}(2,q^{m})$ is a maximal nonlinear affine set of less than q ^m points.

However, if s>2, we conjecture that the maximal set is linear.

Conjecture 21

If $\mathcal {U}$ is a maximal affine set that determines the set $\mathcal {D}$ of directions and t=s>2, then $\mathcal {U}$ is an affine GF(s)-linear set.

Although we conjecture that the maximal affine sets with s=t>2 are linear sets, the converse is not true.

Example

(Nonmaximal affine linear set)

Let AG(2,s ⁱ) be a canonical subgeometry of AG(2,q=s ^i⋅j), and let $\mathcal {U}$ be an affine GF(s)-linear set in the subgeometry AG(2,s ⁱ) that contains more than s ⁱ points. Then $\mathcal {U}$ determines the same direction set that is determined by the subgeometry AG(2,s ⁱ).