An analogue of Ruzsa's conjecture for polynomials over finite fields

Abstract

In 1971, Ruzsa conjectured that if $f:\mathbb{N}\to \mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod k for every $n,k\in \mathbb{N}$ and $f(n)=O({\theta }^n)$ with $\theta <e$ then f is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field using the Polynomial Method in combinatorics.

Introduction

Let $\mathbb{N}$ denote the set of positive integers and let ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$. A strong form of a conjecture by Ruzsa is the following assertion. Suppose that $f:{\mathbb{N}}_0\to \mathbb{Z}$ satisfies the following 2 properties:

• $f(n+p)\equiv f(n)$ mod p for every prime p and every $n\in {\mathbb{N}}_0$;

• $\underset{n\to \infty }{\lim \sup }\frac{\log |f(n)|}{n}<e$.

Then f is necessarily a polynomial. The original form allows the version of (P1) in which p is not necessarily a prime. Hall [11] gave an example constructed by Woodall showing that the upper bound e in (P2) is optimal. The reasoning behind this upper bound as well as the Hall-Woodall example is the (equivalent version of the) Prime Number Theorem stating that the product of primes up to n is $e^{n+o(n)}$ and the fact that the residue class of $f(n)$ modulo this product is determined uniquely by $f(0),\dots ,f(n-1)$ thanks to (P1). In 1971, Hall [10] and Ruzsa [15] independently proved the following result.

Theorem 1.1 Hall-Ruzsa, 1971

Suppose that $f:{\mathbb{N}}_0\to \mathbb{Z}$ satisfies (P1) and

$$\underset{n\to \infty }{\lim \sup }\frac{\log |f(n)|}{n}<e-1$$

then f is a polynomial.

The best upper bound was obtained in 1996 by Zannier [18] by extending earlier work of Perelli and Zannier [17], [13]:

Theorem 1.2 Zannier, 1996

Suppose that $f:{\mathbb{N}}_0\to \mathbb{Z}$ satisfies (P1) and

$$\underset{n\to \infty }{\lim \sup }\frac{\log |f(n)|}{n}<e^{0.75}$$

then f is a polynomial.

In fact, the author remarked [18, pp. 400–401] that the explicit upper bound $e^{0.75}$ was chosen to avoid cumbersome formulas and it was possible to increase it slightly. The method of [18] uses the fact that the generating series $\sum f(n)x^n$ is D-finite over $\mathbb{Q}$ (i.e. it satisfies a linear differential equation with coefficients in $\mathbb{Q}(x)$) [13, Theorem 1.B] then applies deep results on the arithmetic of linear differential equations [4], [6].

This paper is motivated by our recent work on D-finite series [3] and a review of Ruzsa's conjecture. From now on, let $\mathbb{F}$ be the finite field of order q and characteristic p, let $\mathcal{A}=\mathbb{F}[t]$, and let $\mathcal{K}=\mathbb{F}(t)$. We have the usual degree map $\mathrm{deg}:\mathcal{A}\to {\mathbb{N}}_0\cup \{-\infty \}$. A map $f:\mathcal{A}\to \mathcal{A}$ is called a polynomial map if it is given by values on $\mathcal{A}$ of an element of $\mathcal{K}[X]$. For every $n\in {\mathbb{N}}_0$, let ${\mathcal{A}}_n=\{A\in \mathcal{A}:\mathrm{deg}(A)=n\}$, ${\mathcal{A}}_{<n}=\{A\in \mathcal{A}:\mathrm{deg}(A)<n\}$, and ${\mathcal{A}}_{\le n}=\{A\in \mathcal{A}:\mathrm{deg}(A)\le n\}$. Let $\mathcal{P}\subset \mathcal{A}$ be the set of irreducible polynomials; the sets ${\mathcal{P}}_n$, ${\mathcal{P}}_{<n}$, and ${\mathcal{P}}_{\le n}$ are defined similarly. The superscript + is used to denote the subset consisting of all the monic polynomials, for example ${\mathcal{A}}^{+}$, ${\mathcal{A}}_n^{+}$, ${\mathcal{P}}_{\le n}^{+}$, etc. From the well-known identity [14, pp. 8]:

$$\prod _{d|n}\prod _{P\in {\mathcal{P}}_d^{+}}P=t^{q^n}-t$$

we have

$$q^n\le \mathrm{deg}\left(\prod _{P\in {\mathcal{P}}_{\le n}^{+}}P\right)<2q^n$$

for every $n\in \mathbb{N}$. In view of the reasoning behind Ruzsa's conjecture, it is natural to ask the following:

Question 1.3

Let $f:\mathcal{A}\to \mathcal{A}$ satisfy the following 2 properties:

• $f(A+BP)\equiv f(A)\mathrm{mod}P$ for every $A,B\in \mathcal{A}$ and $P\in \mathcal{P}$;

• $\underset{\mathrm{deg}(A)\to \infty }{\lim \sup }\frac{\log \mathrm{deg}(f(A))}{\mathrm{deg}(A)}<q$.

Is it true that f is a polynomial map?

Note that (P3) should be the appropriate analogue of (P1): over the natural numbers, iterating (P1) yields $f(n+bp)\equiv f(n)\mathrm{mod}p$ for every $n,b\in {\mathbb{N}}_0$ and prime p. On the other hand, over $\mathcal{A}$, due to the presence of characteristic p, iterating the congruence condition $f(A+P)\equiv f(A)\mathrm{mod}P$ for $A\in \mathcal{A}$ and $P\in \mathcal{P}$ is not enough to yield (P3). By the following example that is similar to the one by Hall-Woodall, we have that the upper bound q in (P4) cannot be increased. Fix a total order ≺ on $\mathcal{A}$ such that $A\prec B$ whenever $\mathrm{deg}(A)<\mathrm{deg}(B)$. We define $g:\mathcal{A}\to \mathcal{A}$ inductively. First, we assign arbitrary values of g at the constant polynomials. Let $n\in \mathbb{N}$, $B\in {\mathcal{A}}_n$, and assume that we have defined $g(A)$ for every $A\in \mathcal{A}$ with $A\prec B$ such that:

$$g(A)\equiv g(A_1)\mathrm{mod}P\text{for every }A,A_1\prec B\text{ and prime }P|(A-A_1).$$

For every $P\in {\mathcal{P}}_{\le n}^{+}$, let $R_P\in \mathcal{A}$ with $\mathrm{deg}(R_P)<\mathrm{deg}(P)$ such that $B\equiv R_P$ mod P. By the Chinese Remainder Theorem, there exists a unique $R\in \mathcal{A}$ with $\mathrm{deg}(R)<\mathrm{deg}\left(\prod _{P\in {\mathcal{P}}_{\le n}^{+}}P\right)$ such that $R\equiv f(R_P)$ mod P for every $P\in {\mathcal{P}}_{\le n}^{+}$. Then we define

$$g(B):=R+\prod _{P\in {\mathcal{P}}_{\le n}^{+}}P.$$

It is not hard to prove that g satisfies Property (P3) (with g in place of f) and for every $n\in \mathbb{N}$, $B\in {\mathcal{A}}_n$, we have $\mathrm{deg}(g(B))\in [q^n,2q^n)$ by (1). This latter property implies that g cannot be a polynomial map.

Following Professor Ruzsa's suggestion, we have:

Definition 1.4

Property (P3) is called the prime congruence-preserving condition. A map $f:\mathcal{A}\to \mathcal{A}$ with this property is called a prime congruence-preserving map.

Our main result implies the affirmative answer to Question 1.3; in fact we can replace (P4) by the much weaker condition that $\mathrm{deg}(f(A))$ is not too small compared to $\frac{q^{\mathrm{deg}(A)}}{\mathrm{deg}(A)}$:

Theorem 1.5

Let $f:\mathcal{A}\to \mathcal{A}$ be a prime congruence-preserving map such that

$$\mathrm{deg}(f(A))<\frac{q^{\mathrm{deg}(A)}}{27q\mathrm{deg}(A)}\mathit{\text{when }}\mathrm{deg}(A)\mathit{\text{is sufficiently large}}.$$

Then f is a polynomial map.

There is nothing special about the constant $1/(27q)$ in (2) and one can certainly improve it by optimizing the estimates in the proof. It is much more interesting to know if the function $q^{\mathrm{deg}(A)}/\mathrm{deg}(A)$ in (2) can be replaced by a larger function (see Section 4). There are significant differences between Ruzsa's conjecture and Question 1.3 despite the apparent similarities at first sight. Indeed none of the key techniques in the papers [13], [18] seem to be applicable in our situation. Obviously, the crucial result used in [18] that the generating series $\sum f(n)x^n$ is D-finite has no counterpart here. The proof of the main result of [13] relies on a nontrivial linear recurrence relation of the form $c_{d}f(n+d)+\dots +c_{0}f(n)=0$. Over the integers, such a relation will allow one to determine $f(n)$ for every $n\ge d$ once one knows $f(0),\dots ,f(n-1)$. On the other hand, for Question 1.3, while it seems possible to imitate the arguments in [13] to obtain a recurrence relation of the form $c_{d}f(A+B_d)+\dots +c_{0}f(A+B_0)=0$ for $A\in \mathcal{A}$ with $d\in \mathbb{N}$ and $B_0,\dots ,B_d\in \mathcal{A}$, such a relation does not seem as helpful: when $\mathrm{deg}(A)$ is large, one cannot use the relation to relate $f(A)$ to the values of f at smaller degree polynomials. Finally, the technical trick of using the given congruence condition to obtain the vanishing on $[2M_0,(2+ϵ)M_0]$ from the vanishing on $[0,M_0]$ (see [13, pp. 11–12] and [18, pp. 396–397]) does not seem applicable here.

The proof of Theorem 1.5 consists of 2 steps. The first step is to show that the points $(A,f(A))$ for $A\in \mathcal{A}$ belong to an algebraic plane curve over $\mathcal{K}$, then it follows that $\mathrm{deg}(f(A))$ can be bounded above by a linear function in $\mathrm{deg}(A)$. The second step, which might be of independent interest, treats the more general problem in which f is prime congruence-preserving and there exists a special sequence ${(A_n)}_{n\in {\mathbb{N}}_0}$ in $\mathcal{A}$ such that $\mathrm{deg}(f(A_n))$ is bounded above by a linear function in $\mathrm{deg}(A_n)$. Both steps rely on the construction of certain auxiliary polynomials; such a construction has played a fundamental role in diophantine approximation, transcendental number theory, and combinatorics. For examples in number theory, the readers are referred to [1], [12] and the references therein. In combinatorics, the method of constructing polynomials vanishing at certain points has recently been called the Polynomial Method and is the subject of the book [9]. This method has produced surprisingly short and elegant solutions of certain combinatorial problems over finite fields [7], [5], [8].

Acknowledgments. We are grateful to Professor Imre Ruzsa for his interest and helpful suggestions. We wish to thank Dr. Carlo Pagano, Professor Umberto Zannier, and the anonymous referee for the useful comments. J. B is partially supported by the NSERC Discovery Grant RGPIN-2016-03632. K. N. is partially supported by the NSERC Discovery Grant RGPIN-2018-03770, a start-up grant at UCalgary, and the CRC tier-2 research stipend 950-231716.