Bounded height in pencils of subgroups of finite rank (Q6583546)

The paper under review deals with the classical problem of finding explicit, or at least effective, upper bounds for the complexity of algebraic solutions to some polynomial equations, which is measured by some form of height. In this case, the paper deals with equations in which the exponents are allowed to vary within certain sets of rational numbers corresponding to subgroups of finite rank of \(\mathbb{G}_m^r\) that are not necessarily finitely generated.\N\NMore precisely, the authors start from a curve \(\mathcal{C}\) defined over the field of algebraic numbers \(\overline{\mathbb{Q}}\), and let \(\mathbb{F} = \overline{\mathbb{Q}}(\mathcal{C})\). Then, the authors take \(\Gamma \subseteq \mathbb{G}_m^r(\mathbb{F})\) to be a finitely generated subgroup that is constant-free, i.e., such that for every map of algebraic groups \(\phi \colon \mathbb{G}_m^r \to \mathbb{G}_m\) one has that \(\phi(\Gamma) \cap \overline{\mathbb{Q}}^\ast \subseteq \overline{\mathbb{Q}}^\ast_\text{tors}\). Then, the authors consider the division group\N\[\N\Gamma^\text{div} := \{ \boldsymbol{\gamma} \in \mathbb{G}_m^r(\overline{\mathbb{F}}) \colon \text{ exists } n \in \mathbb{N} \ \text{such that} \ \boldsymbol{\gamma}^n \in \Gamma \},\N\]\Nwith the objective to understand the set of pairs \((P,\boldsymbol{\gamma})\) with \(P \in \mathcal{C}(\overline{\mathbb{Q}})\) and \(\boldsymbol{\gamma} \in \Gamma^\text{div}\), such that the specialization \(\boldsymbol{\gamma}_P \in \mathbb{G}_m^r(\overline{\mathbb{Q}})\) is well defined and lies in the specialization \(V_P\), where \(V \subseteq \mathbb{G}_m^r\) is any closed algebraic subvariety defined over \(\mathbb{F}\). Clearly, if \(\boldsymbol{\gamma} \in \Gamma^\text{div} \cap V\) then any \(P \in \mathcal{C}(\overline{\mathbb{Q}})\) yields such a pair \((P,\boldsymbol{\gamma})\), hence we should exclude those \(\boldsymbol{\gamma} \in \Gamma^\text{div} \cap V\) from the analysis.\N\NMoreover, if in the problem considered in the previous paragraph one replaces \(\Gamma^\text{div}\) with \(\Gamma\) itself, then the authors of the paper under review have proven in previous work [Duke Math. J. 166, No. 13, 2599--2642 (2017; Zbl 1431.11084)] that for any point \(P \in \mathcal{C}(\overline{\mathbb{Q}})\) belonging to a pair \((P,\boldsymbol{\gamma})\) as above such that \(\boldsymbol{\gamma} \in \Gamma \setminus V\), the height \(h(P)\), normalized by taking any ``reasonable'' projective model of \(\mathcal{C}\), is bounded from above by an effective quantity depending on \(\mathcal{C}\), \(\Gamma\) and \(V\).\N\NUnfortunately, this relatively simple statement does not generalize easily to the case considered in the paper under review, when one replaces \(\Gamma\) with \(\Gamma^\text{div}\). However, the authors could solve all the problems that arise at least when \(V\) is a linear hypersurface with constant coefficients, given by an equation \(\alpha_1 x_1 + \dots + \alpha_r x_r = 1\), where \(x_1,\dots,x_r\) are the coordinates of \(\mathbb{G}_m^r\) while \(\boldsymbol{\alpha} := (\alpha_1,\dots,\alpha_r) \in \mathbb{G}_m^r(\overline{\mathbb{Q}})\). In this case, the authors prove in Theorem 1.5 of the paper under review that for every real number \(\varepsilon\) such that \(0 < \varepsilon < 1\), there exist a real number \(B_\varepsilon\), depending effectively on \(\mathcal{C}\), \(\Gamma\), \(\boldsymbol{\alpha}\) and \(\varepsilon\), another real number \(C\) depending effectively only on \(\mathcal{C}\) and \(\Gamma\), and a finite set \(H \subseteq \Gamma^\text{div}\) depending effectively only on \(\mathcal{C}\) and \(\Gamma\), such that for any pair \((P,\boldsymbol{\gamma})\) as above either \(h(P)\) is bounded above by \(B_\varepsilon\), or \(h^\mathbb{A}_\text{geo}(\boldsymbol{\gamma}) \leq C\) and there exists \(\boldsymbol{\eta} \in H\) such that \(\mathrm{dist}(\boldsymbol{\eta},\boldsymbol{\gamma}) < \varepsilon\) and \(\boldsymbol{\gamma}_P \boldsymbol{\eta}_P^{-1} \boldsymbol{\eta} \in V\), or there exists \(I \subsetneq \{1,\dots,r\}\) such that \(\sum_{i \in I} \alpha_i (\gamma_i)_P = 0\). Here, \(h^\mathbb{A}_\text{geo}\) is the affine geometric height, defined as\N\[\Nh^\mathbb{A}_\text{geo}(\mathbf{f}) := -\sum_{P \in \mathcal{C}(\overline{\mathbb{Q}})} \min\{0,\mathrm{ord}_P(f_1),\dots,\mathrm{ord}_P(f_r)\}\N\]\Nfor any \(\mathbf{f} = (f_1,\dots,f_r) \in \mathbb{A}^r(\mathbb{F})\), which can be canonically extended to \(\overline{\mathbb{F}}\). Moreover, this height allows one to define a distance on \(\mathbb{G}_m^r(\overline{\mathbb{F}})/\mathbb{G}_m^r(\overline{\mathbb{Q}})\) by setting \(\mathrm{dist}(\boldsymbol{\alpha},\boldsymbol{\beta}) := h^\mathbb{A}_\text{geo}(\alpha_1 \beta_1^{-1}) + \dots + h^\mathbb{A}_\text{geo}(\alpha_r \beta_r^{-1})\).\N\NLet us conclude this review with a couple of remarks. First of all, the three cases appearing in the theorem's conclusion can all occur. This shows in particular that it will likely be challenging to generalize the aforementioned Theorem 1.5 of the paper under review from linear hypersurfaces to other kinds of subvarieties of \(\mathbb{G}_m^r\). Second, the proof of Theorem 1.5 proceeds by a reduction to rank one subgroups when the height of \(\boldsymbol{\gamma}\) is large. On the other hand, if \(\boldsymbol{\gamma}\) has small height then the authors approximate it first with some \(\mathbf{v} \in \Gamma^\text{div}\) having bounded denominator, and then proceed by a careful analysis of the successive minima of the geometric height on the vector space of those \(\mathbf{g} \in \overline{\mathbb{F}}^r\) such that \(\mathbf{g} \cdot \mathbf{v} \in V\). Finally, Theorem 1.5 is already sufficient to treat several explicit examples. For instance, the authors prove that if \(t \in \overline{\mathbb{Q}}\) satisfies \(\alpha_1 t^\lambda + \alpha_2 (1-t)^\lambda = 1\) for some \(\lambda \in \mathbb{Q}_{> 0}\) and some \(\alpha_1,\alpha_2 \in \overline{\mathbb{Q}}^\ast\) such that \((\alpha_1 t^{\lambda - 1}, \alpha_2 (1-t)^{\lambda - 1}) \neq (1,1)\) then \(h(t) \leq 100 \max(1,\lambda^{-1}) + 121 \lambda^{-1} (h(\alpha_1) + h(\alpha_2))\). When \(\alpha_1 = \alpha_2 = 1\) and \(\lambda\) is an integer, this equation was considered by \textit{F. Beukers} [J. Pure Appl. Algebra 117--118, 97--103 (1997; Zbl 0876.11009)]. Moreover, similar considerations apply to the equations \(\alpha_1 t^\lambda + \alpha_2 (1-t)^\mu = 1\) and \(\alpha_1 t^\lambda + \alpha_2 (1-t)^\lambda + \alpha_3(1+t)^\lambda = 1\), where again \(\lambda\) and \(\mu\) are rational numbers and \(\alpha_1, \alpha_2, \alpha_3\) are algebraic numbers.\N\NTo conclude, the paper under review provides a nice investigation into the possibility of bounding effectively the height of families of algebraic equations. These families are certainly quite special, but they will almost surely provide a stepping block to generalize the previous results proven by the authors of the paper under review from finitely generated subgroups to subgroups of finite rank.