A strengthening of McConnel’s theorem on permutations over finite fields

Abstract

Let p be a prime, $q=p^n$, and $D \subset \mathbb {F}_q^*$. A celebrated result of McConnel states that if D is a proper subgroup of $\mathbb {F}_q^*$, and $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel’s theorem extends if D has small doubling.

1 Introduction

Throughout this paper, let p be a prime. Let $q=p^n$ and $\mathbb {F}_q$ be the finite field with q elements with $\mathbb {F}_q^*=\mathbb {F}_q \setminus \{0\}$ .

A celebrated result of McConnel [15] states that if D is a proper subgroup of $\mathbb {F}_q^*$ , and $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that

(1.1) $$ \begin{align} \frac{f(x)-f(y)}{x-y} \in D \end{align} $$

whenever $x,y \in \mathbb {F}_q$ with $x \neq y$ , then there are $a,b \in \mathbb {F}_q$ and an integer $0 \leq j \leq n-1$ , such that $f(x)=a x^{p^j}+b$ for all $x \in \mathbb {F}_q$ . This result was first proved by Carlitz [7] when q is odd and D consists of squares in $\mathbb {F}_q^*$ , that is, D is the subgroup of index $2$ . Carlitz’s theorem and McConnel’s theorem have various connections with finite geometry, graph theory, and group theory; we refer to a nice survey by Jones [11, Section 9]. In particular, they have many applications in finite geometry; see, for example, [5, 12]. We also refer to variations of McConnel’s theorem in [6, 10, 13] via tools from group theory.

One may wonder, if the assumption that D is a multiplicative subgroup plays an important role in McConnel’s theorem and if it is possible to weaken this assumption. Inspired by this natural question, in this paper, we find a sufficient condition on D so that if condition (1.1) holds for a function $f:\mathbb {F}_q \to \mathbb {F}_q$ , then f necessarily has the form $f(x)=a x^{p^j}+b$ . In particular, our main result (Theorem 1.2) strengthens McConnel’s theorem.

Before stating our results, we introduce some motivations and backgrounds from the theory of directions and their applications. Let $AG(2,q)$ denote the affine Galois plane over the finite field $\mathbb {F}_q$ . Let U be a subset of points in $AG(2,q)$ ; we use Cartesian coordinates in $AG(2,q)$ so that $U=\{(x_i,y_i):1 \leq i \leq |U|\}$ . The set of directions determined by $U \subset AG(2, q)$ is

$$\begin{align*}\mathcal{D}_U=\left\{ \frac{y_j-y_i}{x_j-x_i} \colon 1\leq i <j \leq |U| \right \} \subset \mathbb{F}_q \cup \{\infty\},\end{align*}$$

where $\infty $ is the vertical direction. If $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function, we can naturally consider its graph $U(f)=\{(x,f(x)): x \in \mathbb {F}_q\}$ and the set of directions determined by f is $\mathcal {D}_f:=\mathcal {D}_{U(f)}$ . Indeed, $\mathcal {D}_f$ precisely computes the set of slopes of tangent lines joining two points on $U(f)$ . Using this terminology, condition (1.1) is equivalent to $\mathcal {D}_f \subset D$ .

The following well-known result is due to Blokhuis, Ball, Brouwer, Storme, and Szőnyi [3, 4].

Theorem 1.1 Let p be a prime and let $q=p^n$ . Let $f:\mathbb {F}_q \to \mathbb {F}_q$ be a function such that $f(0)=0$ . If $|\mathcal {D}_f|\leq \frac {q+1}{2}$ , then f is a linearized polynomial, that is, there are $\alpha _0,\alpha _1, \ldots , \alpha _{n-1}\in \mathbb {F}_q$ , such that

$$ \begin{align*}f(x)=\sum_{j=0}^{n-1} \alpha_j x^{p^j}, \quad \forall x \in \mathbb{F}_q. \end{align*} $$

In particular, when $q=p$ is a prime, Theorem 1.1 implies McConnel’s theorem; this was first observed by Lovász and Schrijver [14]. We remark that when $q=p$ is a prime, Müller [16] showed a stronger result, namely, if D is a non-empty proper subset of $\mathbb {F}_p^*$ such that $f(x)-f(y)\in D$ whenever $x-y \in D$ , then there are $a,b \in \mathbb {F}_p$ such that $f(x)=ax+b$ for all $x \in \mathbb {F}_p$ . For a general prime power q, Theorem 1.1 does not imply McConnel’s theorem directly. However, using the idea of directions, Muzychuk [17] provided a self-contained proof of McConnel’s theorem.

While Theorem 1.1 is already quite powerful, in many applications, if some extra information on $\mathcal {D}_f$ is given, it is desirable to obtain a stronger conclusion, namely, $f(x)$ is of the form $ax^{p^j}+b$ for some j, or even $f(x)$ must have the form $ax+b$ . As an illustration, we mention two recent works in applying a stronger version of Theorem 1.1 to prove analogs of the Erdős–Ko–Rado (EKR) theorem in the finite field setting [1, 2]. Asgarli and Yip [2] proved the EKR theorem for a family of pseudo-Paley graphs of square order, and their main result roughly states that if $\mathcal {D}_f$ arises from a Cayley graph with “nice multiplicative properties” on its connection set, then $f(x)$ has the form $ax+b$ ; see [19] for more discussions. As another example, very recently, Aguglia, Csajbók, and Weiner proved several EKR theorems for polynomials over finite fields [1]. Again, one key ingredient in their proof is a strengthening of Theorem 1.1. In [1, Theorem 2.2], they showed that if $\mathcal {D}_f$ is a proper $\mathbb {F}_p$ -subspace of $\mathbb {F}_q$ , then $f(x)$ has the form $ax+b$ ; in [1, Theorem 2.13] (see also [9]), they showed that if $\mathcal {D}_f\setminus \{0\}$ is a subset of a coset of K, where K is the subgroup of $\mathbb {F}_q^*$ with index $2$ , then $f(x)$ has the form $ax^{p^j}+b$ for some j.

Inspired by the connection above between the theory of directions and McConnel’s theorem, we establish the following result.

Theorem 1.2 Let p be a prime. Let $q=p^n$ and $D \subset \mathbb {F}_q^*$ . Suppose $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that

$$ \begin{align*}\frac{f(x)-f(y)}{x-y} \in D \end{align*} $$

whenever $x,y \in \mathbb {F}_q$ with $x \neq y$ . If

(1.2) $$ \begin{align} |DD^{-1}D^{-1}|\leq \frac{q+1}{2}, \end{align} $$

then there are $a,b \in \mathbb {F}_q$ and an integer $0 \leq j \leq n-1$ , such that $f(x)=a x^{p^j}+b$ for all $x \in \mathbb {F}_q$ . In particular, if $c>0$ is a real number such that $|DD|\leq c|D|$ , and $c^3|D|\leq \frac {q+1}{2}$ , then the same conclusion holds.

If D is a proper subgroup of $\mathbb {F}_q^*$ , then clearly $DD^{-1}D^{-1}=D$ and thus Theorem 1.2 recovers McConnel’s theorem. Indeed, in this case, the doubling constant of D is simply $|DD|/|D|=1$ . Our result shows that if the doubling constant $|DD|/|D|$ of D is small, and $|D|$ is not too large, then the analog of McConnel’s theorem still holds. There are many ways to construct a set $D \subset \mathbb {F}_q^*$ with small doubling. For example, we can set $D=K \cup E$ , where K is a subgroup of $\mathbb {F}_q^*$ , and E is an arbitrary subset of $\mathbb {F}_q^*$ such that $|E|$ is small; alternatively, D can be taken to be the union of some cosets of a fixed subgroup K of $\mathbb {F}_q^*$ .

For a linearized polynomial $f(x)$ , note that $\mathcal {D}_f=\operatorname {Im}(f(x)/x)$ . We refer to [8] and references therein on the study of $\operatorname {Im}(f(x)/x)$ for linearized polynomials f. In particular, it is an open question to determine all the possible sizes of $\operatorname {Im}(f(x)/x)$ among linearized polynomials f [8, Section 6]. Theorem 1.2 implies the following corollary, which partially addresses this question. It states that if D is such an image set and D satisfies inequality (1.2), then D is necessarily a coset of a subgroup of $\mathbb {F}_{q}^*$ with a restricted index.

Corollary 1.3 Let p be a prime and let $q=p^n$ . If $D \subset \mathbb {F}_q^*$ and $|DD^{-1}D^{-1}|\leq \frac {q+1}{2}$ , then $D=\mathcal {D}_f$ for some function $f: \mathbb {F}_q \to \mathbb {F}_q$ with $f(0)=0$ if and only if $D=aK$ , where $a \in \mathbb {F}_q^*$ and K is a subgroup of $\mathbb {F}_{q}^*$ with index $p^r-1$ , where r is a divisor of n.

Notations. We follow standard notations for arithmetic operations among sets. Given two sets A and B, we write $AB=\{ab: a \in A, b \in B\}$ , $A^{-1}=\{a^{-1}: a \in A\}$ .

2 Proofs

We start by proving Theorem 1.2. Our proof is inspired by several arguments used in [14, 17].

Proof (Proof of Theorem 1.2) Without loss of generality, by replacing the function $f(x)$ with $f(x)-f(0)$ , we may assume that $f(0)=0$ . Since $|\mathcal {D}_f|\leq \frac {q+1}{2}$ , Theorem 1.1 implies that f is linearized. Let $g(x)=1/f(x^{-1})$ for $x \in \mathbb {F}_q \setminus \{0\}$ and set $g(0)=0$ . We claim that g is also linearized.

Let $x,y \in \mathbb {F}_q^*$ with $x \neq y$ . Then,

$$ \begin{align*}\frac{g(x)-g(y)}{x-y}=\frac{\frac{1}{f(x^{-1})}-\frac{1}{f(y^{-1})}}{x-y}=\frac{f(y^{-1})-f(x^{-1})}{(x-y)f(x^{-1})f(y^{-1})}=\frac{f(y^{-1}-x^{-1})}{(x-y)f(x^{-1})f(y^{-1})} \end{align*} $$

since f is linearized. It follows that

$$ \begin{align*}\frac{g(x)-g(y)}{x-y}=\frac{f(\frac{x-y}{xy})}{(x-y)f(x^{-1})f(y^{-1})}=\frac{f(\frac{x-y}{xy})}{\frac{x-y}{xy}} \cdot \frac{x^{-1}}{f(x^{-1})} \cdot \frac{y^{-1}}{f(y^{-1})} \in DD^{-1}D^{-1}. \end{align*} $$

On the other hand, if $x \in \mathbb {F}_q^*$ , then

$$ \begin{align*}\frac{g(x)-g(0)}{x-0}=\frac{g(x)}{x}=\frac{1}{xf(x^{-1})}=\frac{x^{-1}}{f(x^{-1})} \in D^{-1}. \end{align*} $$

We conclude that

$$ \begin{align*}\mathcal{D}_g \subset DD^{-1}D^{-1} \cup D^{-1}=DD^{-1}D^{-1}. \end{align*} $$

Then inequality (1.2) implies that $|\mathcal {D}_g|\leq \frac {q+1}{2}$ , and Theorem 1.1 implies that g is also linearized.

Since f and g are both linearized, we can find $\alpha _0, \beta _0, \ldots , \alpha _{n-1}, \beta _{n-1} \in \mathbb {F}_q$ , such that

$$ \begin{align*}f(x)=\sum_{j=0}^{n-1} \alpha_j x^{p^j}, \quad \text{and} \quad g(x)=\sum_{j=0}^{n-1} \beta_j x^{p^j}, \quad \forall x \in \mathbb{F}_q. \end{align*} $$

By the definition of g, we have $f(x^{-1})g(x)=1$ for each $x \in \mathbb {F}_q^*$ , and thus $(x^{p^{n-1}}f(x^{-1})) (g(x)/x)=x^{p^{n-1}-1}$ for all $x \in \mathbb {F}_q^*$ . Equivalently,

(2.1) $$ \begin{align} h(x):=\bigg(\sum_{j=0}^{n-1} \alpha_j x^{p^{n-1}-p^j}\bigg) \bigg(\sum_{j=0}^{n-1} \beta_j x^{p^j-1} \bigg)=x^{p^{n-1}-1} \end{align} $$

holds for all $x \in \mathbb {F}_q^*$ . Note that $h(x)-x^{p^{n-1}-1}$ is a polynomial with degree at most $2(p^{n-1}-1)\leq p^n-1=q-1$ , and $h(x)-x^{p^{n-1}-1}$ vanishes on $\mathbb {F}_q^*$ . It follows that there is a constant $C \in \mathbb {F}_q$ , such that $h(x)-x^{p^{n-1}-1}=C(x^{q-1}-1)$ as polynomials. By setting $x=0$ , we get $C=0$ . Therefore, $h(x)=x^{p^{n-1}-1}$ as polynomials. In particular, $g(x)$ is a factor of $x^{p^{n-1}}$ . Thus, there are $\gamma \in \mathbb {F}_q^*$ and $0 \leq j \leq n-1$ such that $g(x)=\gamma x^{p^j}$ for all $x \in \mathbb {F}_q$ , and it follows that $f(x)=\gamma ^{-1} x^{p^j}$ for all $x \in \mathbb {F}_q$ , as required.

Finally, assume that $|DD|\leq c|D|$ , and $c^3|D|\leq \frac {q+1}{2}$ . Then the Plünnecke–Ruzsa inequality (see, for example, [18, Theorem 1.2]) implies that

$$ \begin{align*}|DD^{-1}D^{-1}|\leq c^3|D| \leq \frac{q+1}{2}, \end{align*} $$

and thus the same conclusion holds.

Now we use Theorem 1.2 to deduce Corollary 1.3.

Proof (Proof of Corollary 1.3) Assume that $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that $f(0)=0$ and $\mathcal {D}_f=D$ . Since $D \subset \mathbb {F}_q^*$ and $|DD^{-1}D^{-1}|\leq \frac {q+1}{2}$ , Theorem 1.2 implies that there exist $a \in \mathbb {F}_q^*$ and an integer $0 \leq j \leq n-1$ , such that $f(x)=ax^{p^j}$ for all $x \in \mathbb {F}_q$ . Therefore, $D=\mathcal {D}_f=\{ax^{p^j-1}: x \in \mathbb {F}_q^*\}.$ Note that

$$ \begin{align*}\gcd(p^j-1, q-1)=\gcd(p^j-1, p^n-1)=p^{\gcd(j,n)}-1.\end{align*} $$

It follows that

$$ \begin{align*}D=\{ax^{p^{\gcd(j,n)}-1}: x \in \mathbb{F}_q^*\}=aK,\end{align*} $$

where K is the subgroup of $\mathbb {F}_q^*$ with index $p^{\gcd (j,n)}-1$ .

Conversely, let r be a divisor of n and $a \in \mathbb {F}_q^*$ . Let $f(x)=ax^{p^r}$ for all $x \in \mathbb {F}_q$ . Then we have $\mathcal {D}_f=\{ax^{p^r-1}: x \in \mathbb {F}_q^*\}=aK$ , where K is the subgroup of $\mathbb {F}_q^*$ with index $p^{r}-1$ .