1 Introduction

Let p be a prime. The points of the projective line \(PG(1,{\mathbb F}_p)\) can be considered as equivalence classes of the non-zero vectors of the affine plane \(AG(2,{\mathbb F}_p)\), where \({\mathbb F}_p\) denotes the field of p elements. Two elements are equivalent if one of them is a non-zero multiple of the other. For a subset H, the set of directions \(D(H) \subset PG(1,{\mathbb F}_p)\) is the equivalence classes corresponding to the elements of \((H-H)\setminus \{ 0\}\). The number of directions determined by H is the cardinality of D(H).

Rédei investigated the number of directions determined by a p-element subset of the 2 dimensional vector space \({\mathbb F}_p^2\) over the finite field \({\mathbb F}_p\) of p elements. Using his results on lacunary polynomials, Rédei proved that such a subset is a line or determines at least \(\frac{p+1}{2}\) directions. Later, Megyesi excluded the case of \(\frac{p+1}{2}\) directions. These results can be summarized as follows, see [7, Theorem 24’ on page 226].

Theorem 1.1

If a set of p points in \({\mathbb F}_p^2\) is not a line, then it determines at least \(\frac{p+3}{2}\) directions.

Sets determining exactly \(\frac{p+3}{2}\) directions were described by Lovász and Schrijver [6]. As a generalization of Rédei’s result, Szőnyi [8] proved that if \(k \le p\), then a k-element subset not lying in a line determines at least \(\frac{k+3}{2}\) directions. Gács [4] showed that there is another gap in the possible number of directions between \(\frac{p+3}{2}\) and \(\lfloor 2 \frac{p-1}{3}\rfloor +1\).

Note that the result of Megyesi and Rédei was independently obtained by Dress, Klin, and Muzychuk [2]. They used this result to give a new proof of Burnside’s theorem on permutation groups of prime degree. Another application of the results on the number of directions in group theory is due to Dona [3], who used his result to add to the theory of growth in groups. Further, the connection of the set of directions in the affine plane to blocking sets on finite projective planes are also discussed in [4].

One of the main purposes of this paper is to give a new proof of Theorem 1.1. The other one is to prove Theorem 1.2, which will immediately imply Theorem 1.1.

Let \(g :{\mathbb F}_p \rightarrow {\mathbb F}_p\) be a polynomial. Considering the polynomial function corresponding to g, we may assume that the degree of g is at most \(p-1\). Further the elements of \({\mathbb F}_p\) can be considered as elements in \(\{0,1, \ldots , p-1 \} \subset \mathbb {Z}\). Thus we may consider the sum of the values in \(\mathbb {Z}\). If it is small enough, then we obtain restrictions on the degree of g.

In order to motivate the following theorem, it is useful to consider the polynomial \(q(x)=x^{\frac{p-1}{2}}+1\) when p is an odd prime. The sum of the values of q is equal to p since

$$\begin{aligned} q(x)= {\left\{ \begin{array}{ll} 2 &{} \text{ if } \lambda \text{ is } \text{ a } \text{ quadratic } \text{ residue },\\ 1 &{} \text{ if } x=0, \\ 0 &{}\text{ otherwise. } \end{array}\right. } \end{aligned}$$
(1)

This simple example shows that the following theorem is sharp.

Theorem 1.2

Let p be an odd prime. If \(\sum _{i=0}^{p-1} g(i) =p \), then either the degree of g is at least \(\frac{p-1}{2}\) or g is a constant function.

2 Technique

The proof of Theorem 1.2 relies on the following result proved in [5]. Note that the proof of this lemma uses Rédei’s polynomials.

The following lemma is formulated for special directions, which is a generalization of directions. For sets of cardinality p, the two definitions coincide. We will only use the notion direction in this paper so we refer the interested reader to [5] for the definition of special directions.

Lemma 2.1

Let A be a subset of \({\mathbb F}_p^2\) of cardinality kp. Assume that it determines \(d \ge 2\) special directions. Let r be a projection function defined as follows:

$$\begin{aligned} r(i)=| \{ j \in {\mathbb F}_p \mid (i,j) \in A \} |. \end{aligned}$$

Notice that r can be uniquely identified with a polynomial of degree at most \(p-1\). Then \(d \ge {\text {deg}}(r)+2 \).

Using suitable affine transformation, we may prove the previous lemma for any projection function obtained in this way instead of the vertical projection.

As a corollary of this lemma, we obtain that, in order to prove Rédei’s result, it is sufficient to prove Theorem 1.2 which is of independent interest.

Using a simple argument, we get a weaker result than Theorem 1.2.

Proposition 2.2

Let p be an odd prime. If \(\sum _{i=0}^{p-1} g(i) =p \), then either the degree of g is at least \(\frac{p-1}{3}\) or g is a constant function.

Proof

We will simply prove that one of the values of g is taken at least \(\frac{p-1}{3}\) times. More precisely \(|\{ x \in {\mathbb F}_p \mid g(x)=0\}|\ge \frac{p-1}{3}\) or \(|\{ x \in {\mathbb F}_p \mid g(x)=1\}|\ge \frac{p-1}{3}\).

Assume indirectly this is not the case. Then

$$\begin{aligned} \sum _{x \in \mathbb {F}_p} g(x) \ge \sum _{x \in \mathbb {F}_p :g(x) \ge 1} 1 + \sum _{x \in \mathbb {F}_p :g(x) \ge 2} 1 \ge \bigg (p-\frac{p-1}{3}\bigg ) + \bigg (p- 2 \frac{p-1}{3}\bigg )=p+1, \end{aligned}$$

a contradiction. \(\square \)

In order to emphasize the usefulness of Lemma 2.1, we prove the following simple result. The proof uses again the observation that the multiplicity of any element in the range of a non-constant polynomial is a lower bound for the degree of a non-zero polynomial.

Theorem 2.3

Let H be a subset of \({\mathbb F}_p^2\) of cardinality p. Let a and b be the size of the projection of H to the x and y axis, respectively. Then the number of directions determined by H is at least \(p-\min \{ a,b\}+2\).

Proof

Let r be the function from \({\mathbb F}_p\) to \({\mathbb F}_p\) defined as in Lemma 2.1. Then the multiplicity of 0 as a root of r is at least \(p-a\). In a similar manner, we may project the set H along the horizontal lines. The corresponding projection polynomial is of degree at least \(p-b\). Now, Lemma 2.1 gives the result. \(\square \)

The importance of this trivial corollary of Lemma 2.1 relies on the similarity of this result to the one of Di Benedetto, Solymosi, and White [1], who proved that the number of directions determined by a subset of \({\mathbb F}_p^2\), which is the Cartesian product of the subsets \(A, B \subset {\mathbb F}_p\) with \(|A|,|B| \ge 2\) and \(|A||B| <p\) is at least

$$\begin{aligned} |A| \cdot |B|- \min \{|A|,|B| \} +2. \end{aligned}$$

This result gives an upper bound for the clique number of the Paley graph.

3 Proof of the main result

Proof of Theorem 1.2

As we have mentioned, every polynomial function from \({\mathbb F}_p\) to \({\mathbb F}_p\) coincides with a unique polynomial of degree at most \(p-1\) so we will automatically reduce the degree below p.

The proof of Theorem 1.2 relies on the following simple observation. The degree of a polynomial h is smaller than \(p-1\) if and only if

$$\begin{aligned} \sum _{y \in {\mathbb F}_p} h(y) \equiv 0 \pmod {p}. \end{aligned}$$

Let us consider the polynomial \(f(x)=g(x^2)\) (reduced to degree at most \(p-1\)). Clearly, if \(\sum _{y \in {\mathbb F}_p} f(y) \not \equiv 0 \pmod {p}\), then \(\text {deg}(g) \ge \frac{p-1}{2}\).

We argue that if there were a non-constant polynomial of degree less than \(\frac{p-1}{2}\) such that the sum of its values is p, then there would be one which takes value 0 at 0. It is clear that if the polynomial is non-constant, then the sum can only be p if 0 is in the range of the polynomial. Now applying a linear substitution \(x \rightarrow x+i\) on the x variable of the polynomial, we obtain a polynomial of the same degree satisfying \(f(0)=0\).

Let us first estimate the sum of the values of f from above.

$$\begin{aligned} \begin{aligned} \sum _{y \in {\mathbb F}_p} f(y)&=\sum _{x \in {\mathbb F}_p} g(x^2)=g(0) + 2 \sum _{x \in ({\mathbb F}_p^*)^2} g(x) \\&= 2 \sum _{x \in ({\mathbb F}_p^*)^2} g(x) \le 2 \sum _{x \in {\mathbb F}_p} g(x)=2p. \end{aligned} \end{aligned}$$
(2)

It is clear that \(\sum _{y \in {\mathbb F}_p} f(y)\) cannot be equal to p since it is an even number by equation (2).

On the other hand, equality in (2) can only hold if g vanishes on all the non-quadratic residues, in which case the degree of g is at least \(\frac{p-1}{2}+1=\frac{p+1}{2}\).

It could be that the previous sum is 0 but then g vanishes on the quadratic residues, again having many roots.

Therefore we obtain that the sum in equation (2) is not divisible by p, finishing the proof of Theorem 1.2. \(\square \)

Theorem 1.2 can be applied to the projection function defined in Lemma 2.1. We may assume that the set is not a vertical line. It follows from Theorem 1.2 that the degree of r is at least \(\frac{p-1}{2}\). Then by Lemma 2.1, the number of directions determined by A is at least \(\frac{p+3}{2}=\frac{p-1}{2}+2\).

There are natural problems arising here. Can we find a similar result proving the ones listed in the beginning of this paper?

  • Is it true that up to affine transformations \(x^{\frac{p-1}{2}}+1\) is the unique polynomial of degree \(\frac{p-1}{2}\) such that the sum of its values is p?

  • Is it possible to find a new proof for Gács’s result on the number of directions using the polynomial method presented in this paper?