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Hermite-Thue equation: Padé approximations and Siegel’s lemma. (English) Zbl 1436.11084
In Diophantine approximation the use of Padé approximants is used to proving sharp transcendence measures. Specifically so-called twin approximations can successfully used to prove Baker-type transcendence measures for \(e\).
Let \(l_1,\ldots,l_m\) be positive integers, let \(\alpha_1,\ldots,\alpha_m\) be distinct variables. Denote \(\overline{\alpha}=(\alpha_1,\ldots,\alpha_m)^{T}\), \(\overline{l}=(l_1,\ldots,l_m)^{T}\) and \(L=l_1+\ldots+l_m\). The twin problem\/ for the \(m\)-tuple of exponentials \(e^{\alpha_j t}\}_{j=1}^m\) is then given by:
Find an explicit polynomial \(B_{\overline{l},0}(t,\overline{\alpha})\), polynomials \(B_{\overline{l},j}(t,\overline{\alpha})\) and remainders \(S_{\overline{l},j}(t,\overline{\alpha}),\ j=1,\ldots,m\) satisfying \[ B_{\overline{l},0}(t,\overline{\alpha}) e^{\alpha_j t}-B_{\overline{l},j}(t,\overline{\alpha}) = S_{\overline{l},j}(t,\overline{\alpha}),\ j=1,\ldots,m \] and \[ \operatorname{deg}B_{\overline{l},j}(t,\overline{\alpha})\leq L, 0\leq j\leq m;\ L+l_j+1\leq \operatorname{ord} S_{\overline{l},j}(t,\overline{\alpha})<\infty,\ 1\leq j\leq m. \]
Moreover, the authors use the wild version of the tame, where the \(l_j\) and \(\overline{l}\), used in the definition of the \(B\)’s, are replaced by \(\nu_j\) and \(\overline{\nu}\): \[ \overline{\nu}=(\nu_1,\ldots,\nu_m)^{T}\in\mathbb{Z}_{\geq 1}^m,\,\nu_1\leq l_1,\ldots,\nu_m\leq l_m, \nu_1+\ldots+\nu_m=:M\leq L \] The main results read as follows:
Theorem 2.1. Let \(\alpha_1,\ldots,\alpha_m\in\mathbb{Z}\). Then \[ \left(\prod_{1\leq j\leq m}\,a_j^{\frac{\nu_j}{2}}\right)\,\prod_{1\leq i < j\leq m}\, (a_i-a_j)^{\min(\nu_i^2,\nu_j^2)}\large{|}_{\mathbb{Z}} D(\overline{a}), \] where \(D(\overline{\alpha})\) is the greatest common divisor of all the \(M\times M\) minors of the matrix \(\mathcal{V}(\overline{a})\in\mathcal{M}_{M\times (L+1)}(\mathbb{Z})\).
Theorem 2.2. Let \(l_1,\ldots,l_m\) be positive integers and let \(\alpha_1,\ldots,\alpha_m\) be distinct variables. Denote \(\overline{\alpha}=(\alpha_1,\ldots,\alpha_m)^{T}\), \(\overline{l}=(l_1,\ldots,l_m)^{T}\) and \(L=l_1+\ldots+l_m\). Then the twin problem for the exponentials, defined above, has a non-zero solution of polynomials \(B_{\overline{l},j}\in\mathbb{Q}[t,\overline{\alpha}]\). Moreover \[ B_{\overline{l}, 0}(t,\overline{\alpha})=\sum_{i=0}^{L}\,\frac{L!}{i!}\,\tau_i(\overline{l},\overline{\alpha}) t^i,\ \tau_i(\overline{l},\overline{\alpha})=\frac{(-1)^i \mathcal{V}[i]}{T(\overline{l},\overline{\alpha})}\,\in\mathbb{Z}[\overline{\alpha}], \] where \[ T(\overline{l},\overline{\alpha}):=\alpha_1^{\frac{l_1}{2}}\cdots \alpha_m^{\frac{\alpha_m}{2}}\,\prod_{1\leq i<j\leq m}\,(\alpha_i-\alpha_j)^{\min\{l_i^2, l_j^2\}} \] is a common factor of the \(L\times L\) minors \(\mathcal{V}[i],\,i=0,\ldots, L\), of the matrix \(\mathcal{V}(\overline{\alpha})\).
The layout of the paper is as follows:
§1. Introduction
1.1 Hermite-Padé approximation
1.2 The twin problem
1.3 Siegel’s lemma
§2. Results
The Theorems 2.1 and 2.2
§3. Preliminaries and tools
3.1 Exterior algebras
3.1.1 Increasing lists
3.1.2 Increasing vectors
3.1.3 Grassmann coordinates
3.2 Generalised minor expansions
3.3 Polynomial rings
§4. Genralised Vandermonde-type polynomial block matrices
4.1 Case A
4.2 Case B
§5. Hermite-Padé approximations to the exponential function: tame case
5.1 A new proof of Theorem 5.1
5.1.1 Factors of \(\mathcal{U}[0]\)
5.1.2 Rank
5.1.3 Cramer’s rule
§6. Hermite-Padé approximations to the exponential function: wild case
6.1 The twin problem
6.2 Siegel’s lemma
6.3 The Bombieri-Vaaler version of Siegel’s lemma
6.4 Common factor
Leads to the proof of Theorem 2.1
6.5 Rank
6.6 Twin type II Padé approximants
Leads to the proof of Theorem 2.2
Appendix A. Some examples of the matrix \(\mathcal{V} (\overline{\alpha})\)
References (\(26\) items)
MSC:
11J81 Transcendence (general theory)
41A21 Padé approximation
11C20 Matrices, determinants in number theory
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