On Pisot's d-th root conjecture for function fields and related GCD estimates

Abstract

We propose a function-field analog of Pisot's d-th root conjecture on linear recurrences, and prove it under some “non-triviality” assumption. Besides a recent result of Pasten-Wang on Büchi's d-th power problem, our main tool, which is also developed in this paper, is a function-field analog of a GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such a GCD estimate, we also obtain an asymptotic result.

Introduction

Let $R(X)={\sum }_{n=0}^{\infty }b(n)X^n$ represent a rational function in $\mathbb{Q}(X)$ and suppose that $b(n)$ is a d-th power in $\mathbb{Q}$ for all large $n\in \mathbb{N}$. Pisot's d-th root conjecture states that one can choose a d-th root $a(n)$ of $b(n)$ such that $\tilde{R}(X):=\sum a(n)X^n$ is again a rational function in $\bar{\mathbb{Q}}(X)$, where we denote by $\bar{k}$ the algebraic closure of a field k. The sequence $\{b(n)\}$ coming from the rational function $R(X)$ is a linear recurrence sequence, which can be written as an exponential polynomial, which we now define. An exponential polynomial over a field k is a sequence $b:\mathbb{N}\to k$ of the form

$$n\mapsto \sum _{i\in I}{B}_i(n){\beta }_i^n,$$

where I is a (finite) set of indices, each ${\beta }_i\in k^{⁎}$ is nonzero and each $B_i\in k[T]$ is a single-variate polynomial. When it can be arranged so that each $B_i$ is constant, we say that b is simple. For any exponential polynomial $b:\mathbb{N}\to k$ and any natural number d, it is clear that $b^d:\mathbb{N}\to k$, defined by $n\mapsto b{(n)}^d$, is still an exponential polynomial. The following result of Zannier [18] essentially proves its converse, which is a generalization of Pisot's d-th root conjecture stated earlier. We also refer to [18] for a survey on related works, and [9] for a more general statement.

Theorem 1 [18]

Let b be an exponential polynomial over a number field k, and $d\ge 2$ be an integer. Suppose that $b(n)$ is the d-th power of some element in k for all but finitely many n. Then there exists an exponential polynomial a over $\bar{k}$ such that $a{(n)}^d=b(n)$ for all n.

The first goal of this paper is to investigate an analog of Theorem 1 over a function field K over an algebraically closed field k with characteristic zero. It will be noticed that our analog is not covered in various existing extensions from Theorem 1, crucially because d-th roots of those $b(n)$ are assumed to exist in K, which is not finitely generated over $\mathbb{Q}$.

Let C be a smooth projective algebraic curve of genus $\mathfrak{g}$ defined over k with its function field equal to K. We will always denote by $\mathfrak{p}$ a point in $C(\mathbf{k})$, by S a finite subset of $C(\mathbf{k})$. Since K contains the algebraically-closed field k, note that for any pair of exponential polynomials $a:\mathbb{N}\to K$ and $c:\mathbb{N}\to \mathbf{k}$ we have that $c{a}^d:\mathbb{N}\to K$, defined by $n\mapsto c(n)a(n){}^d$, is still an exponential polynomial whose n-th term is the d-th power of some element in K for all $n\in \mathbb{N}$. A plausible statement obtained from Theorem 1 by replacing the number field k by our function field K must therefore have its conclusion modified to the existence of exponential polynomials a, c, respectively over $\bar{K}$ and over k, such that $c(n)a{(n)}^d=b(n)$ for all n. Our result in this direction is as follows.

Theorem 2

Let $b(n)={\sum }_{i=1}^{\ell }{B}_i(n){\beta }_i^n$ be an exponential polynomial over K, i.e. $B_i\in K[T]$ and ${\beta }_i\in K^{⁎}$. Let Γ be the multiplicative subgroup of $K^{⁎}$ generated by ${\beta }_1,\dots ,{\beta }_{\ell }$. Let L be the smallest finite extension over K such that all the ${\beta }_i$ and the leading coefficient of ${\sum }_{i=1}^{\ell }{B}_i(T)$ have d-th roots in L. Assume that $\mathit{\Gamma }\cap {\mathbf{k}}^{⁎}=\{1\}$. If $b(n)$ is a d-th power in K for infinitely many $n\in \mathbb{N}$, then there exists an exponential polynomial $a(m)={\sum }_{i=1}^r{A}_i(m){\alpha }_i^m$, $A_i\in L[T]$, ${\alpha }_i\in {\bar{L}}^{⁎}$ and a polynomial $R\in \mathbf{k}[T]$ such that $b(m)=R(m)a{(m)}^d$ for all $m\in \mathbb{N}$.

Remark 3

The assumption that $b(n)$ is a d-th power in K for infinitely many $n\in \mathbb{N}$ is weaker than the one in Theorem 1. We refer to [5] for a result over number field under both this weaker assumption and the existence of a dominant root ${\beta }_i$, i.e. a unique ${\beta }_i$ of maximal or minimal absolute value. However, the similar problem on the quotient of two linear recurrences is solved [6] under this weaker assumption without assuming the existence of a dominant root.

Remark 4

In the notation of Theorem 2, it is standard to notice that b consists of the q disjoint subsequences $b_j$, defined by $n\mapsto b(j+qn)$, where $j\in \{0,\dots ,q-1\}$ and q is the order of the torsion subgroup of Γ; moreover, each $b_j$ is an exponential polynomial whose associated ${\mathrm{\Gamma }}_j$ is torsion-free. With this observation, we may generalize Theorem 2 so that the assumption $\mathrm{\Gamma }\cap {\mathbf{k}}^{⁎}=\{1\}$ is relaxed to that $\mathrm{\Gamma }\cap {\mathbf{k}}^{⁎}$ is finite and the conclusion only holds for some $b_j$ rather than b.

Remark 5

In the case where b is simple, i.e., each $B_i$ is constant, we can relax the hypothesis on Γ in Theorem 2 so that the case where $\mathrm{\Gamma }\cap {\mathbf{k}}^{⁎}$ is infinitely cyclic generated by γ can be also treated. In this new case, modifying slightly our proof of Theorem 2, we can conclude that $b(m)=c(m)a{(m)}^d$ for all $m\in \mathbb{N}$, where a is an exponential polynomial over an explicitly-described finite extension $L^{\prime }$ of K, and c is a simple exponential polynomial over k given by $m\mapsto {\sum }_{i=1}^r{c}_i{\gamma }^{e_{i}m}$ for some $c_i\in \mathbf{k}$ and $e_i\in \mathbb{Z}$; but the description of $L^{\prime }$ is different from that of L given in Theorem 2. It seems difficult to further relax the hypothesis on Γ.

Our proof of Theorem 2 contains two major ingredients, both rely on the special features of function fields of characteristic zero. One of the ingredients is the result (restated as Theorem 23 in Section 4) by Pasten and Wang [15] motivated by Büchi's d-th power problem, which has a similar flavor as Pisot's d-th root conjecture but arising from different purposes. While working on an undecidability problem related to Hilbert's tenth problem in the 1970s, Büchi formulated a related arithmetic problem, which can be stated in more generality as follows: Let k be a number field. Does there exist an integer M such that the only monic polynomials $G\in k[T]$ of degree d satisfying that $G(1),\dots ,G(M)$ are d-th powers in k, are precisely those of the form $G(X)={(X+c)}^d$ for some $c\in k$. This problem remains unsolved, while its analogs have been investigated intensively in the recent years. In particular, the analog over function fields of characteristic zero was solved completely, even with an explicit bound on M; see [2] and [14]. We refer to [15] for a survey of relevant works. The other ingredient, which is also developed in this paper, is the function-field analog of the recent work of Levin [10] for number fields and Levin-Wang [11] for meromorphic functions on GCD estimates of two multivariable polynomials over function fields evaluated at arguments which are S-units. We will use the estimates through the following result, which is of independent interest.

Theorem 6

Let $d\ge 2$ be an integer and $F\in K[x_1,\dots ,x_n]$. Assume that F can not be expressed as $a{\mathbf{x}}^{\mathbf{i}}{G}^d$ for any $a\in K^{⁎}$, any monomial ${\mathbf{x}}^{\mathbf{i}}\in K[x_1,\dots ,x_n]$, and any $G\in K[x_1,\dots ,x_n]$. Then we have the following conclusion: For any $u_1,\dots ,u_n\in {\mathcal{O}}_S^{⁎}$, there exists positive integer m and rationals $c_1$ and $c_2$ all depending only on $(d,n,\mathrm{deg}F,{\max }_{1\le j\le n}h(u_j))$, such that if $F(u_1^{\ell },\dots ,u_n^{\ell })$ is a d-th power in K with some $\ell \ge c_1\tilde{h}(F)+c_2\max \{1,2\mathfrak{g}-2+|S|\}$, then $u_1^{m_1}\cdots u_n^{m_n}\in \mathbf{k}$ for some $(m_1,\dots ,m_n)\in {\mathbb{Z}}^n\setminus \{(0,\dots ,0)\}$ with $\sum |m_i|\le 2m$.

Here $\tilde{h}(F)$ is the relevant height of F to be defined in the next section.

Remark 7

We cannot drop a in the assumption of Theorem 6. For example, if $a=u_1\in {\mathcal{O}}_S^{⁎}$ and $F(x_1):=a{x}_1$, we always have that $F(u_1^{d\ell -1})$ is a d-th power in K for all $\ell \in \mathbb{N}$.

The assumption in Theorem 2 that $\mathrm{\Gamma }\cap {\mathbf{k}}^{⁎}$ is trivial implies that every minimal set of generators of Γ is multiplicatively independent modulo k. This suggests how Theorem 6 plays a role in our proof of Theorem 2.

We briefly describe the core idea connecting GCD estimates and our proof of Theorem 6, as introduced by Corvaja-Zannier [7]. After some reduction, we only need to treat the case where F is d-th power free. Given a tuple $(u_1,\dots ,u_n,y)\in {({\mathcal{O}}_S^{⁎})}^n\times K$ which satisfies that $y^d=F(u_1,\dots ,u_n)$, we will construct a polynomial $G\in K[x_1,\dots ,x_n]$ with controllable height, depending on F and the $\frac{u_i^{\prime }}{u_i}$, such that ${(y^d)}^{\prime }=G(u_1,\dots ,u_n)$, where ′ denotes a global derivation on K. For example, if $F:=x_1^2+\cdots +x_n^2$, then our construction will yield $G:=2\frac{u_1^{\prime }}{u_1}{x}_1^2+\dots +2\frac{u_n^{\prime }}{u_n}{x}_n^2$. As $d\ge 2$, the number of common zeros of $y^d$ and ${(y^d)}^{\prime }$ is essentially larger than the number of zeros of $y^{d-1}$. On the other hand, we expect the number of common zeros of $F(u_1,\dots ,u_n)$ and $G(u_1,\dots ,u_n)$ to be essentially smaller than the number of zeros of $y^{d-1}$ unless something special happens. To formalize this idea, we prove the following result on GCD estimates, where all notation involved are defined in Section 2.

Theorem 8

Let $S\subset C$ be a finite set of points. Let $F,G\in K[x_1,\dots ,x_n]$ be a coprime pair of nonconstant polynomials. For any $ϵ>0$, there exist an integer m, positive reals $c_i$, $0\le i\le 4$, all depending only on ϵ, such that for all n-tuple $(g_1,\dots ,g_n)\in {({\mathcal{O}}_S^{⁎})}^n$ with

$$\underset{1\le i\le n}{\max }h(g_i)\ge c_1(\tilde{h}(F)+\tilde{h}(G))+c_2\max \{0,2\mathfrak{g}-2+|S|\},$$

we have that either

$$h(g_1^{m_1}\cdots g_n^{m_n})\le c_3(\tilde{h}(F)+\tilde{h}(G))+c_4\max \{0,2\mathfrak{g}-2+|S|\}$$

holds for some integers $m_1,\dots ,m_n$, not all zeros with $\sum |m_i|\le 2m$, or the following two statements hold.

• $N_{S,\gcd }(F(g_1,\dots ,g_n),G(g_1,\dots ,g_n))\le ϵ{\max }_{1\le i\le n}h(g_i)$;

• $h_{\gcd }(F(g_1,\dots ,g_n),G(g_1,\dots ,g_n))\le ϵ{\max }_{1\le i\le n}h(g_i)$ if we further assume that not both of F and G vanish at $(0,\dots ,0)$.

Remark 9

In Theorem 8, all the quantities claimed to exist can be given effectively. Moreover, the explicit bounds on heights are important in our application. As said earlier, if we are given $F=x_1^2+\cdots +x_n^2$ in Theorem 6, then in the main step of the proof, we construct $G_{\mathbf{u}}:=2\frac{u_1^{\prime }}{u_1}{x}_1^2+\dots +2\frac{u_n^{\prime }}{u_n}{x}_n^2$ for each tuple $\mathbf{u}:=(u_1,\dots ,u_n)\in {({\mathcal{O}}_S^{⁎})}^n$ and apply Theorem 8 to estimate the GCD of $F(\mathbf{u})$ and $G_{\mathbf{u}}(\mathbf{u})$. The main point which makes the proof of Theorem 6 work is that $\tilde{h}(G_{\mathbf{u}})$ can be explicitly bounded independent of these u. (See Proposition 17.)

It is more desirable to obtain GCD estimates, such as Statement (a) and (b) in Theorem 8, under the assumption that $g_1,\dots ,g_n$ are multiplicatively independent modulo k. As a result in this direction, we can actually replace the right hand side of (1.2) by 0 in the case where $n=2$ and the coefficients of F and G are in k. We include a complete statement below. Although this result can be deduced from [7, Corollary 2.3], we will derive it from our proof of Theorem 8.

Theorem 10

Let $F,G\in \mathbf{k}[x_1,x_2]$ be nonconstant coprime polynomials. For any $ϵ>0$, there exist an integer m, constant c, both depending only on ϵ, such that for all pairs $(g_1,g_2)\in {({\mathcal{O}}_S^{⁎})}^2$ with $\max \{h(g_1),h(g_2)\}\ge c\max \{1,2\mathfrak{g}-2+|S|\}$, either we have that $g_1^{m_1}{g}_2^{m_2}\in \mathbf{k}$ holds for some integers $m_1,m_2$, not all zeros with $|m_1|+|m_2|\le 2m$, or the following two statements hold

• $N_{S,\gcd }(F(g_1,g_2),G(g_1,g_2))\le ϵ\max \{h(g_1),h(g_2)\}$;

• $h_{\gcd }(F(g_1,g_2),G(g_1,g_2))\le ϵ\max \{h(g_1),h(g_2)\}$, if we further assume that not both of F and G vanish at $(0,0)$.

We also refer to [4] for a gcd theorem for a special case with $C={\mathbb{P}}^1$ and $S=\{0,\infty \}$. As another result in the same direction, we obtain easily from Theorem 8 that an effectively asymptotic version of Statement (a) and (b) in Theorem 8 holds, merely assuming that $g_1,\dots ,g_n$ are multiplicatively independent modulo k; here the effectivity means that we have an effective lower bound for ℓ in the following statement.

Theorem 11

Let $F,G\in K[x_1,\dots ,x_n]$ be nonconstant coprime polynomials. Let $g_1,\dots ,g_n\in K^{⁎}$, not all constant. Then for any $ϵ>0$, there exist an integer m and constant $c_1$ and $c_2$ depending only on ϵ, such that for each positive integer

$$\ell >c_1(\tilde{h}(F)+\tilde{h}(G))+c_2(\mathfrak{g}+n\underset{1\le i\le n}{\max }\{h(g_i)\}),$$

either we have $g_1^{m_1}\cdots g_n^{m_n}\in \mathbf{k}$ for some integers $m_1,\dots ,m_n$, not all zeros with $\sum |m_i|\le 2m$, or the following two statements hold.

• $N_{S,\gcd }(F(g_1^{\ell },\dots ,g_n^{\ell }),G(g_1^{\ell },\dots ,g_n^{\ell }))\le ϵ{\max }_{1\le i\le n}h(g_i^{\ell })$;

• $h_{\gcd }(F(g_1^{\ell },\dots ,g_n^{\ell }),G(g_1^{\ell },\dots ,g_n^{\ell }))\le ϵ{\max }_{1\le i\le n}h(g_i^{\ell })$, if we further assume that not both of F and G vanish at $(0,\dots ,0)$.

Remark 12

When $F,G\in \mathbb{C}[x_1,\dots ,x_n]$ be a coprime pair of nonconstant polynomials and $g_1,\dots ,g_n\in \mathbb{C}[z]$ are multiplicatively independent modulo $\mathbb{C}$, then the results in [11] also imply Statement (a) and (b) in Theorem 11. Our statement here is stronger since we have formulated effective bounds on ℓ and the $m_i$ such that $g_1^{m_1}\cdots g_n^{m_n}\in \mathbf{k}$. When $n>2$, the only other previous result in this direction appears to be a result of Ostafe [12, Th. 1.3], which considers special polynomials such as $F=x_1\cdots x_r-1,G=x_{r+1}\cdots x_n-1$, but proves a uniform bound in place of Statement (a) and (b) independent of ℓ. In the case where $n=2$, previous results include the original theorem of Ailon-Rudnick [1] in this setting, i.e. $F=x_1-1$, $G=x_2-1$, and extensions of Ostafe [12] and Pakovich and Shparlinski [13] (all with uniform bounds). It is noted in [12] that it appears to be difficult to extend the techniques used there to obtain results for general F and G.

We collect the background materials in Section 2. We will prove some main lemmas in Section 3. The proofs of Theorem 2 and Theorem 6 are given in Section 5 and Section 4 respectively. Finally, we establish the gcd theorems in Section 6.