Multiplicative dependence of rational values modulo approximate finitely generated groups (Q6644158)

Let \(\mathbb{C}^{*}\) be the set of all non-zero complex numbers. We say that a collection of complex numbers \(\alpha_1, \alpha_2, \ldots, \alpha_n \in \mathbb{C}^{*}\) are \textit{multiplicatively dependent} if there exists integers \(k_1, k_2, \ldots, k_n\), not all zero, such that\N\[\N\alpha_1^{k_1} \alpha_2^{k_2} \cdots \alpha_n^{k_n} = 1.\N\]\NMore generally, given any subset \(G \subseteq \mathbb{C}^{*}\), we say that \(\alpha_1, \alpha_2, \ldots, \alpha_n \in \mathbb{C}^{*}\) are \textit{multiplicatively dependent modulo \(G\)} if there exists integers \(k_1, k_2, \ldots, k_n\), not all zero, such that\N\[\N\alpha_1^{k_1} \alpha_2^{k_2} \cdots \alpha_n^{k_n} \in G.\N\]\NFor instance, people have been interested in multiplicative dependence of algebraic numbers.\N\NLet \(K\) be a number field, \(\overline{K}\) be an algebraic closure of \(K\), \(K^*\) be the unit subgroup of \(K\), and \(\Gamma\) be a finitely generated subgroup of \(K^*\). For any \(A \subseteq K^*\), define\N\[\NA^{\text{div}} := \{ \alpha \in \overline{K} \, : \, \alpha^m \in A \text{ for some integer } m \ge 1 \}.\N\]\NFor any \(\epsilon > 0\) and finitely generated subgroup \(\Gamma \subseteq K^*\), define its \textit{approximate division subgroup} by\N\[\N\Gamma_\epsilon^{\text{div}} := \{ \alpha \beta \, : \, \alpha \in \Gamma^{\text{div}}, \, \beta \in \overline{K} - \{ 0 \} \text{ with } h(\beta) \le \epsilon \}\N\]\Nwhere \(h( \cdot )\) is the absolute logarithmic Weil height function. In this paper under review, the authors are interested in studying multiplicative dependence modulo \(\Gamma_\epsilon^{\text{div}}\) among values of polynomials or rational functions over \(K\). For example, the following results are proved.\N\begin{itemize}\N\item[1.] Let \(f_1, f_2, \ldots, f_n \in K[X]\) be pairwise coprime polynomials. Assume each of them has at least two distinct roots. Then, for every \(\epsilon > 0\), there are only finitely many elements \(\alpha \in K\) such that \(f_1(\alpha), \ldots, f_n(\alpha)\) are multiplicatively dependent modulo \(\Gamma_\epsilon^{\text{div}}\).\N\item[2.] Let \(f_1, f_2, \ldots, f_n \in K(X)\) be non-constant rational functions such that they are multiplicatively independent modulo constants. Assume that, for each \(f_i\), its numerator either has no linear factor or has at least two distinct linear factor and so does its denominator. Assume further than \(f_1, \ldots, f_n\) have distinct linear factors over \(K\) (if they have any). Then, for every \(\epsilon > 0\), there are only finitely many elements \(\alpha \in K\) such that \(f_1(\alpha), \ldots, f_n(\alpha)\) are multiplicatively dependent modulo \(\Gamma_\epsilon^{\text{div}}\).\N\item[3.] If \(f_1, f_2 \in K[X]\) be polynomials of degree at least \(2\) with at least two distinct roots. Assume that they cannot multiplicatively generate a power of a linear fractional function. The, for any \(\epsilon > 0\), there are only finitely many elements \(\alpha \in K\) such that \(f_1(\alpha)\) and \(f_2(\alpha)\) are multiplicatively dependent modulo \(\Gamma_\epsilon^{\text{div}}\).\N\end{itemize}\N\NThe authors point out that the conditions ``polynomials being coprime'' and ``having at least two distinct roots'' are necessary. The following are a few key ingredients being used:\N\begin{itemize}\N\item Northcott's Theorem that the set \(\mathcal{A}(K, H) = \{ \alpha \in K^* \, : \, h(\alpha) \le H \}\) is finite.\N\N\item A result of [\textit{A. Ostafe} and \textit{I. E. Shparlinski}, in: Number theory -- Diophantine problems, uniform distribution and applications. Festschrift in honour of Robert F. Tichy's 60th birthday. Cham: Springer. 347--368 (2017; Zbl 1393.37113)]\N\[\NK^* \cap \Gamma_\epsilon^{\text{div}} \subseteq \{ \beta \eta \, : \, (\beta, \eta) \in \Gamma \times \mathcal{A}(K, \epsilon + r H) \}\N\]\Nwhere \(\Gamma\) is generated by \(\{g_1, \ldots, g_r \}\) with \(H = \max_{i = 1, \ldots, r} h(g_i)\).\N\N\item A finiteness result of [\textit{A. Bérczes} et al., Int. Math. Res. Not. 2021, No. 12, 9045--9082 (2021; Zbl 1490.37115)] on the set \(\{ \alpha \in K : f(\alpha) \in \Gamma \}\) for rational functions \(f \in K(X)\) not of the form \(a (X - b)^d\) or \(a (X - b)^d / (X - c)^d\)\N\N\item A finiteness result of \textit{G. Maurin} [Int. Math. Res. Not. 2011, No. 23, 5259--5366 (2011; Zbl 1239.14020)] concerning the union of all algebraic subgroups in \(\mathbb{G}_m^n\) of codimension at least \(2\) where \(\mathbb{G}_m = \mathbb{C}^{*}\) endowed with the multiplicative group law.\N\N\item A result of \textit{R. Tijdeman} and \textit{A. Schinzel} [Acta Arith. 31, 199--204 (1976; Zbl 0303.10016)] on the finiteness of the number of solutions to hyper-elliptic equation \(y^m = P(x)\) (particularly used in the proof of the above third stated result).\N\end{itemize}\N\NThe authors also study multiplicative dependence among \(n\)th compositional iterate \(\underbrace{f \circ f \circ \cdots \circ f}_{n \text{ times}}\) of a rational function \(f \in K(X)\) using a result of \textit{M. Young} [Monatsh. Math. 192, No. 1, 225--247 (2020; Zbl 1442.11166)]. They also highlight another motivation that the study of intersection of varieties in \(\mathbb{G}_m^n\) with sets of the type \(\Gamma_\epsilon^{\text{div}}\) falls within two famous conjectures:\N\begin{itemize}\N\item[1.] Mordell-Lang conjecture on intersection of varieties with finitely generated subgroups\N\item[2.] Bogomolov conjecture on the discreteness of the set of points of bounded height in a variety\N\end{itemize}\Nand post the general open question on what conditions the following set\N\[\N\{ \alpha \in K( \Gamma^{\text{div}}) \, : \, f_1(\alpha), \ldots, f_n(\alpha) \text{ multiplicatively dependent mod } \Gamma^{\text{div}} \}\N\]\Nis finite.