Unlikely intersections over finite fields: polynomial orbits in small subgroups

TL;DR

This paper investigates how often polynomial iterations land in small subgroups of finite fields and provides bounds on subgroup sizes generated by polynomial orbits, advancing understanding of polynomial dynamics over finite fields.

Contribution

It introduces new bounds on polynomial orbit intersections with subgroups and extends results to sequences of polynomial compositions over finite fields.

Findings

Estimates the frequency of polynomial orbit intersections with subgroups.

Provides lower bounds on subgroup sizes generated by polynomial orbits.

Develops general results for sequences of polynomial compositions.

Abstract

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.

Equations

$$u_0=u\text{mbox}and{u}_{n+1}=f(u_n),n=1,2,\dots$$

$$u_n\in \mathcal{G}\cup \{0\},n=1,\dots ,N.$$

$${\mathcal{V}}_u(N)=\{f_{i_1}\circ \dots \circ f_{i_n}(u):\text{leavevmode}{i}_1,\dots ,i_n\in \{1,\dots ,k\},0\le n\le N\}$$

$$V_u(N)=\#{\mathcal{V}}_u(N)$$

$${\mathcal{V}}_u(N)\subseteq \mathcal{G}\cup \{0\}.$$

$$G_k(N)⩾V_u(N).$$

$$G_k(N)\ge c_2\min \{V_u(N{)}^{2-\epsilon },p^{1-\epsilon }\}\text{for }N\ge 1.$$

$$G(N)\ge c_2\min \left\{N^{2-\epsilon },p^{1-\epsilon }\right\}.$$

$$T_k(N,\mathcal{G})=\#({\mathcal{V}}_u(N)\cap \mathcal{G})$$

$${\rho }_k(N)=\frac{T_k(N,\mathcal{G})}{V_u(N)}.$$

$$V_u(N{)}^{\epsilon }<\#\mathcal{G}<\min \{V_u(N{)}^{2-\epsilon },p^{1-\epsilon }\},$$

$${\rho }_k(N)<c_2\frac{1}{\log V_u(N)}.$$

$$N^{\epsilon }<\#\mathcal{G}<\min \{N^{2-\epsilon },p^{1-\epsilon }\},$$

$$\rho (N)<c_2\frac{1}{\log N}.$$

$$L_N(u,\mathcal{A};{\omega }_1,\dots ,{\omega }_{\ell })=\#\{v$$

$$d(u,{\omega }_i(v))\le N,{\omega }_i(v)\in \mathcal{A},i=1,\dots ,\ell \}.$$

$$B(k,h)=\left\{\begin{array}[]{cl}h&\text{if }k=1,\\ \frac{k^{h}-1}{k-1}&\text{otherwise}.\end{array}\right.$$

$$\#\{v\in \mathcal{A}:\text{leavevmode}d$$

$$\ge \max \left\{3B(k,h),\frac{3\ell }{h}\#\left\{v\in \mathcal{V}(\mathcal{H}):d(u,v)\le N\right\}\right\},$$

$$L_N(u,\mathcal{A};{\omega }_1,\dots ,{\omega }_{\ell })\gg \frac{h}{B(k,h{)}^{\ell +1}}\#\{v\in \mathcal{A}:d(u,v)\le N\},$$

$$\beta =B(k,h)\text{mbox}and\nu =\#\{v\in \mathcal{V}(\mathcal{H}):d(u,v)\le N\}.$$

$$d(u,z)\le N-hd(z,v)\le h.$$

$$P\ge h(\#\{v\in \mathcal{A}:\text{leavevmode}d(u,v)\le N\}-\beta )$$

$$\#\{v\in \mathcal{A}:d(z,v)\le h\}\ge \ell .$$

$$P\le \beta J+(\ell -1)(\nu -J).$$

$$J$$

$$\gg \frac{h}{\beta }\#\{v\in \mathcal{A}:d(u,v)\le N\}.$$

$$\left(\genfrac{}{}{0ex}{}{\beta }{\ell }\right)⩽{\beta }^{\ell }$$

$$L_N(u,\mathcal{A};{\omega }_1,\dots ,{\omega }_k)$$

$$\gg \frac{h}{{\beta }^{\ell +1}}\#\{v\in \mathcal{A}:d(u,v)\le N\}$$

$$g_i(x_i)=0\text{but}{g}_j(x_i)\ne 0\text{for}0\le i,j\le k,i\ne j.$$

$$g_{i_1}\circ \dots \circ g_{i_r},r\le h,i_1,\dots ,i_r\in \{1,\dots ,\ell \}$$

Full text

1. Introduction

1.1. Background and motivation

Recently, several of so called unlikely intersection type results, see [28] for a general background, have been obtained on the scarcity of elements in orbits of polynomial maps in fields of characteristic zero that fall in a set of prescribed additive, multiplicative or algebraic structure. Examples of such sets include

• •

algebraic varieties [1, 18, 24, 26, 27] where the problem is also known as the dynamical Mordell-Lang conjecture;

• •

an orbit generated by another polynomial or rational function [9, 10];

• •

the set of all roots of unity in $\mathbb{C}$, see [8, 15] and more generally, of algebraic numbers with all conjugates bounded by some constant, see [6, 20];

• •

the set of all perfect powers in a number field, see [3, 17];

• •

a finitely generated group in a number field, see [2, 11, 19].

As we have mentioned, the above results are all established for fields of characteristic zero. There are also some analogues, or sometimes even stronger results for function fields of positive characteristic, where some additional tools are available, such as a very strong (and rigorously established) form of the $abc$-conjecture (see [12]).

On the other hand, there are very few results in this direction in the settings of finite fields. In fact, the problem itself has to be reformulated and instead of asking for finiteness results (which is immediate in this case) one can ask about upper bounds on the size of these intersections compared to the orbit length. For example, multiplicative character sums along elements of very long orbits have been estimated in [13] (using the ideas of [14] one can apparently improve the result of [13] and make it nontrivial for slightly shorter orbits). In turn, this immediately allows to study intersections of orbits with multiplicative subgroups of finite fields. Some of the results of [22, 23] apply to very short segments of an orbit, however they are not uniform with respect to the polynomials $f$ defining the dynamical system, that is, the bounds depend on the size of the coefficients of $f$, which is not quite natural in the settings of finite fields. It is interesting to note that relatively more is known about additive properties of orbits in finite fields, where a range of new techniques becomes available, see [7, 4, 5, 16, 21].

The main goal of this work is to bridge this gap and present some results which are both nontrivial to short segments of orbits and uniform with respect to the polynomial $f$.

Additionally we consider a more general case, when instead of one polynomial, finitely many polynomials are composed with each other. In other words, we consider orbits of a semigroup generated by a finite set of polynomials over a finite field.

Our technique is based on a very recent result of Vyugin [25] which is based on a rather delicate blend of techniques and ideas coming from additive combinatorics and the so-called Stepanov method, which is of an algebraic nature.

1.2. Set-up

Here, motivated by the results outline in Section 1.1, we consider analogous problems in positive characteristic and in particular investigate the frequency of values in an orbit of a polynomial over a finite field which fall in a multiplicative subgroup of a given order.

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements. Given a polynomial $f\in\mathbb{F}_{q}[X]$, we consider the trajectories generated by iterations of $f$ starting from some $u\in\mathbb{F}_{q}$, that is, sequences of the form

[TABLE]

Clearly, each trajectory is eventually periodic. That is, there are some integers $0\leq s<t\leq q$ such that $u_{s}=u_{t}$ and thus $u_{s+n}=u_{t+n}$, $n=0,1,\ldots$ In particular, the smallest $T_{u}=t$ satisfying the above condition is called the trajectory length.

For $u\in\mathbb{F}_{q}$ let $\mathscr{P}_{u}$ be the period length of the sequence (1.1) with starting point $u$, that is, the minimal $k>0$ such that $u_{n}=u_{n+k}$ for $n=0,1,\ldots$ with the convention, that $\mathscr{P}_{u}=\infty$ if the sequence (1.1) is not periodic (just eventually periodic).

Here we consider the size $G(N)$ of the smallest subgroup ${\mathcal{G}}\subseteq\mathbb{F}_{q}^{*}$ which contains the first $N$ non-zero elements of the sequence $(u_{n})$, that is,

[TABLE]

Some bounds on $G(N)$ and also on the frequency of the event $u_{n}\in{\mathcal{G}}$ for a given subgroup ${\mathcal{G}}\subseteq\mathbb{F}_{q}^{*}$, and related questions, are given in [22, 23]. However, the results of [22] apply to either very long segments of an orbit or (as well as in [23] are not uniform with respect to $f$ (that is, depend of the size of the coefficients of $f$). Here we use a different approach, utilizing a recent result of Vyugin [25] and obtain stronger and fully uniform results. In fact, we study analogous problems in a much more general situation of semigroups generated by several polynomials under composition.

Let $f_{1},\ldots,f_{k}\in\mathbb{F}_{q}[X]$ be polynomials of positive degree. Consider the functional graph ${\mathcal{H}}(\mathbb{F}_{q})$ with vertices $\mathbb{F}_{q}$ and directed edges $(x,y)$ with $f_{i}(x)=y$, $1\leq i\leq k$.

For a point $u\in\mathbb{F}_{q}$ we consider the vertices in this graph which are close to $u$:

[TABLE]

and study whether these points are contained in small subgroups. Put

[TABLE]

and define $G_{k}(N)$ as the size of the smallest subgroup ${\mathcal{G}}\subseteq\mathbb{F}_{q}^{*}$ which contains all non-zero elements of ${\mathcal{V}}_{u}(N)$:

[TABLE]

1.3. Results

In order to state the results, for a graph ${\mathcal{H}}(\mathbb{F}_{q})$ we denote by $\overline{{\mathcal{H}}}(\mathbb{F}_{q})$ the undirected graph (that is, $(x,y)$ is an edge in $\overline{{\mathcal{H}}}(\mathbb{F}_{q})$ if $(x,y)$ or $(y,x)$ is an edge in ${\mathcal{H}}(\mathbb{F}_{q})$) and we also denote by $\mathscr{C}_{0}$ the length of a shortest cycle in the graph $\overline{{\mathcal{H}}}(\overline{\mathbb{F}}_{q})$ (defined over the algebraic closure $\overline{\mathbb{F}}_{q}$ of $\overline{\mathbb{F}}_{q}$) which contains [math]. Clearly, for the case of one polynomial ($k=1$), we have $\mathscr{C}_{0}=\mathscr{P}_{0}$.

We remark that the size (or finiteness) of $\mathscr{C}_{0}$ seems to reappear in many works on unlikely intersections of orbits (in finite or infinite fields), see [13, 17], however there seems to be no intrinsic reason for this. It is highly desirable to gain better understanding of the phenomenon.

Since these underlying results of [25] apply only in prime fields, we limit our considerations to prime finite fields $\mathbb{F}_{p}$ where $p$ is a large prime. All explicit and implicit constants are independent of $p$.

We now show that $G_{k}(N)$ grows at least quadratically compared to $V_{u}(N)$ as long as the field size $p$ is large enough to accommodate this growth, improving the trivial linear bound

[TABLE]

Theorem 1.1**.**

Let $\varepsilon>0$, $k\geq 1$ and $d\geq 2$ be fixed. Then there exist constants $c_{1}$ and $c_{2}$ depending only on $d$, $k$ and $\varepsilon$, such that if $f_{1},\ldots,f_{k}\in\mathbb{F}_{p}[X]$ are polynomials of degree at most $d$ and $\mathscr{C}_{0}\geq c_{1}$, then

[TABLE]

As a special case, we have the following result for the dynamical system (1.1) defined by just one polynomial.

Corollary 1.2**.**

For any $\varepsilon>0$ and $d\geq 2$ there exist constants $c_{1}$ and $c_{2}$ depending only on $d$ and $\varepsilon$ such that if $f\in\mathbb{F}_{p}[X]$ is of degree at most $d$, $\mathscr{P}_{0}\geq c_{1}$ and $N<T_{u}$, where $T_{u}$ is the trajectory length of $(u_{n})$, then

[TABLE]

We now show that for subgroups ${\mathcal{G}}$ of size smaller than $G_{k}(N)$ the frequency of elements of the orbit ${\mathcal{V}}_{u}(N)$ which fall in ${\mathcal{G}}$ is small. More precisely, for a subgroup ${\mathcal{G}}\subseteq\mathbb{F}_{p}^{*}$ we denote

[TABLE]

and next give an upper bound of the frequency

[TABLE]

Theorem 1.3**.**

For any $\varepsilon>0$, $d\geq 2$ and $k\geq 1$ there exist positive constants $c_{1}$ and $c_{2}$, depending only on $k,d$ and $\varepsilon$, such that if $f_{1},\dots,f_{k}\in\mathbb{F}_{p}[X]$ are of degree at most $d$, ${\mathcal{G}}$ is a subgroup of $\mathbb{F}_{p}^{*}$ with

[TABLE]

then for $2\leqslant V_{u}(N)<d^{c_{1}\mathscr{C}_{0}}$ we have

[TABLE]

As a special case, we have the following result for the dynamical system (1.1) defined by just one polynomial.

Corollary 1.4**.**

For any $\varepsilon>0$ and $d\geq 2$ there exist positive constants $c_{1}$ and $c_{2}$, depending only on $d$ and $\varepsilon$, such that if $f\in\mathbb{F}_{p}[X]$ is of degree $d$, ${\mathcal{G}}$ is a subgroup of $\mathbb{F}_{p}^{*}$ with

[TABLE]

then for $2\leqslant N<\min\{d^{c_{1}\mathscr{P}_{0}},\,T_{u}\}$ we have

[TABLE]

2. Auxiliary results

2.1. Notation

We use the Landau symbol $O$ and the Vinogradov symbol $\ll$. Recall that the assertions $U=O(V)$ and $U\ll V$ are both equivalent to the inequality $|U|\leqslant cV$ with some constant $c>0$, which throughout this paper, may depend on $k$ and $\varepsilon$, but is independent of the prime $p$ and polynomials $f_{1},\ldots,f_{k}\in\mathbb{F}_{q}[X]$.

For a finite set ${\mathcal{S}}\subseteq\mathbb{Z}$, it is convenient to define ${\mathcal{S}}^{\infty}$ as the set of all finite words in the alphabet ${\mathcal{S}}$.

2.2. Dense sets in graphs

Here we present a graph theory result which can be of independent interest.

Let ${\mathcal{H}}$ be a directed graph, with possible multiple edges. Let ${\mathcal{V}}({\mathcal{H}})$ be the set of vertices of ${\mathcal{H}}$. For $u,v\in{\mathcal{V}}({\mathcal{H}})$, let $d(u,v)$ be the distance from $u$ to $v$, that is, the length of a shortest (directed) path from $u$ to $v$. Assume, that all the vertices have the out-degree $k\geq 1$, and the edges from all vertices are labeled by $\{1,\dots,k\}$.

For a word $\omega\in\{1,\dots,k\}^{\infty}$ over the alphabet $\{1,\dots,k\}$ and $u\in{\mathcal{V}}({\mathcal{H}})$, let $\omega(u)\in{\mathcal{V}}({\mathcal{H}})$ be the end point of the walk started from $u$ and following the edges according to $\omega$.

Let us fix $u\in{\mathcal{V}}({\mathcal{H}})$ and a subset of vertices ${\mathcal{A}}\subseteq{\mathcal{V}}({\mathcal{H}})$. Then for words $\omega_{1},\dots,\omega_{\ell}$ put

[TABLE]

We need the following combinatorial statement which we believe can find further applications in the study of semigroups generated by several polynomials or other functions.

To state the results, for $k,h\geq 1$ let $B(k,h)$ denote the size of the complete $k$-tree of depth $h-1$, that is

[TABLE]

Lemma 2.1**.**

Let $u\in{\mathcal{V}}({\mathcal{H}})$, and $h,\ell\geq 1$ be fixed. If ${\mathcal{A}}\subseteq{\mathcal{V}}({\mathcal{H}})$ is a subset of vertices with

[TABLE]

then there exist words $\omega_{1},\ldots,\omega_{\ell}\in\{1,\dots,k\}^{\infty}$ of length at most $h$ such that

[TABLE]

where the implied constant depends only on $\ell$.

Proof.

Put

[TABLE]

Let $P$ be the number of pairs $(z,v)\in{\mathcal{V}}({\mathcal{H}})\times{\mathcal{A}}$ such that

[TABLE]

If $v\in{\mathcal{A}}$ and $h\leq d(u,v)\leq N$, then there is a path from $u$ to $v$ of length at least $h$, thus there are at least $h$ vertices $z\in{\mathcal{V}}({\mathcal{H}})$ such that $d(u,z)\leq N-h$ and $d(z,v)\leq h$. As there are at most $\beta$ vertices $v$ with $d(u,v)<h$, we have

[TABLE]

On the other hand, let $J$ be the number of $z\in{\mathcal{V}}({\mathcal{H}})$ such that $d(u,z)\leq N-h$ and

[TABLE]

Then

[TABLE]

Comparing (2.2) and (2.3), we derive

[TABLE]

There are at most

[TABLE]

choices for $\omega_{1},\ldots,\omega_{\ell}$, so we see that there is at least one choice, that

[TABLE]

which concludes the proof. ∎

2.3. Polynomial values of subgroups

We call the set of polynomials $g_{1},\ldots,g_{k}\in\mathbb{F}_{p}[X]$ admissible if there exist $x_{1},\ldots,x_{k}\in\overline{\mathbb{F}}_{p}$ such that

[TABLE]

Lemma 2.2**.**

For polynomials $g_{1},\ldots,g_{\ell}\in\mathbb{F}_{p}[X]$, if $h<\mathscr{C}_{0}/2$, then the polynomials

[TABLE]

are admissible.

Proof.

For a given $F$ of form $F=g_{i_{1}}\circ\ldots\circ g_{i_{r}}$, $r\leq h$, $i_{1},\ldots,i_{r}\in\{1,\ldots,\ell\}$, let $x\in\overline{\mathbb{F}}_{p}$ be a zero of $F$. If it is also a zero of $G$ of form $G=g_{j_{1}}\circ\ldots\circ g_{j_{s}}$, $s\leq h$, $j_{1},\ldots,j_{r}\in\{1,\ldots,\ell\}$, $G\neq F$, then there are two different paths from $x$ to 0 in the functional graph ${\mathcal{H}}(\overline{\mathbb{F}}_{p})$ along the edges $i_{r},\dots,i_{1}$ and $j_{s},\dots,j_{1}$. As $r,s\leq h$ these two paths define a cycle of size at most $r+s\leq 2s<\mathscr{C}_{0}$, a contradiction. ∎

Our main tool is the following result of Vyugin [25].

Lemma 2.3**.**

Let ${\mathcal{G}}$ be a subgroup of $\mathbb{F}_{p}^{*}$ and let ${\mathcal{G}}_{1},\ldots,{\mathcal{G}}_{k}$ be $k\geqslant 2$ cosets of ${\mathcal{G}}$. If $g_{1},\ldots,g_{k}\in\mathbb{F}_{p}[X]$ form an admissible set of polynomials of degrees $m_{1}\leq\ldots\leq m_{k}$ and

[TABLE]

then

[TABLE]

where for $\mathbf{m}=(m_{1},\ldots,m_{k})$ we define

[TABLE]

and

[TABLE]

3. Proof of main results

3.1. Proof of Theorem 1.1

Let $h\geq 1$ be the smallest integer such that

[TABLE]

where $B(k,h)$ is defined by (2.1). For an appropriate choice of the constant $c_{1}$, we can assume, that $h<\mathscr{C}_{0}/2$.

We also set

[TABLE]

Let ${\mathcal{G}}$ be the group generated by ${\mathcal{V}}_{u}(N)$. Then

[TABLE]

for

[TABLE]

Moreover, the polynomials

[TABLE]

are admissible by Lemma 2.2.

For an appropriate choice of the constant $c_{1}$ we have

[TABLE]

We can also assume, that

[TABLE]

as otherwise there is nothing to prove. Then by Lemma 2.3

[TABLE]

Finally, we remark that

[TABLE]

which gives the desired result.

3.2. Proof of Theorem 1.3

We can assume throughout the proof, that both $p$ and $N$ are large enough.

Define

[TABLE]

Put

[TABLE]

Clearly, $0<\delta_{0}<1/2$. By the above choice of parameters, we have

[TABLE]

Assume, that

[TABLE]

Then we have

[TABLE]

By Lemma 2.1, there are $\omega_{1},\ldots,\omega_{\ell}\in\{1,\dots,k\}^{\infty}$ of form $\omega_{i}=(\omega_{i,1},\ldots,\omega_{i,r_{i}})$, $r_{i}\leq h$, $1\leq i\leq\ell$ such that

[TABLE]

Consider the polynomials $F_{i}=f_{\omega_{i,1}}\circ\ldots\circ f_{\omega_{i,r_{i}}}$ of degree $m_{i}=\deg F_{i}\leq d^{h}$, $i=1,\ldots,\ell$.

If ${\mathcal{G}}$ is large enough in terms of $\ell$, then

[TABLE]

Moreover, if $\ell$ is large enough, then

[TABLE]

By the choice of $h$,

[TABLE]

by an appropriate choice of $c_{1}$. Then the polynomials $F_{1},\ldots,F_{\ell}$ are admissible by Lemma 2.2. Then by Lemma 2.3 we have

[TABLE]

Comparing (3.2) and (3.3), we have

[TABLE]

Using $\ell<4\varepsilon^{-1}-2$ we see then the exponent of $V_{u}(N)$ is negative, that is

[TABLE]

which contradicts (3.1), provided that $V_{u}(N)$ is large enough.

Acknowledgement

The research of L.M. was supported by the Austrian Science Fund (FWF): Project P31762, and of I.S. was supported in part by the Australian Research Council Grants DP170100786 and DP180100201.