Hermite-Thue equation: Padé approximations and Siegel’s lemma. (English) Zbl 1436.11084

Matala-aho, Tapani; Seppälä, Louna

J. Number Theory 191, 345-383 (2018).

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)

Reviewer: Marcel G. de Bruin (Heemstede)

MSC:

11J81	Transcendence (general theory)
41A21	Padé approximation
11C20	Matrices, determinants in number theory

Keywords:

Diophantine approximation; Padé approximation; Siegel’s lemma; Vandermonde-type determinant

References:

[1]	Baker, A., On some Diophantine inequalities involving the exponential function, Canad. J. Math., 17, 616-626, (1965) · Zbl 0147.30901
[2]	Baker, A., Transcendental number theory, (1975), Cambridge University Press · Zbl 0297.10013
[3]	Baker, G.; Graves-Morris, P. R., Padé approximants, (1996), Cambridge University Press · Zbl 0923.41001
[4]	Bombieri, E.; Vaaler, J. D., On Siegel’s lemma, Invent. Math., 73, 11-32, (1983) · Zbl 0533.10030
[5]	de Bruin, M. G., Some aspects of simultaneous rational approximation, (Numerical Analysis and Mathematical Modelling, Banach Center Publ., vol. 24, (1990), PWN Warsaw), 51-84
[6]	Ernvall-Hytönen, A-M.; Leppälä, K.; Matala-aho, T., An explicit Baker-type lower bound of exponential values, Proc. Roy. Soc. Edinburgh Sect. A, 145, 1153-1182, (2015) · Zbl 1387.11059
[7]	Ernvall-Hytönen, A-M.; Matala-aho, T.; Seppälä, L., On Mahler’s transcendence measure for e, Constr. Approx., (2018)
[8]	Fel’dman, N. I.; Nesterenko, Yu. V., Number theory IV, Encyclopaedia Math. Sci., vol. 44, (1998), Springer Berlin
[9]	Flowe, R. P.; Harris, G. A., A note on generalized Vandermonde determinants, SIAM J. Matrix Anal. Appl., 14, 1146-1151, (1993) · Zbl 0789.15004
[10]	Fukshansky, L., Siegel’s lemma with additional conditions, J. Number Theory, 120, 13-25, (2006) · Zbl 1192.11018
[11]	Hata, M., Remarks on Mahler’s transcendence measure for e, J. Number Theory, 54, 81-92, (1995) · Zbl 0839.11027
[12]	Hermite, C., Sur la fonction exponentielle, C. R. Acad. Sci. Paris, 77, (1873) · JFM 05.0248.01
[13]	Huttner, M., Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions, Israel J. Math., 153, 1-43, (2006) · Zbl 1143.34057
[14]	Krattenthaler, C., Advanced determinant calculus, The Andrews Festschrift, Maratea, 1998, Sém. Lothar. Combin., 42, (1999), Art. B42q, 67 pp. · Zbl 0923.05007
[15]	Mahler, K., On a paper by A. Baker on the approximation of rational powers of e, Acta Arith., 27, 61-87, (1975) · Zbl 0305.10030
[16]	Matala-aho, T., Type II Hermite-Padé approximations of generalized hypergeometric series, Constr. Approx., 33, 289-312, (2011) · Zbl 1236.41017
[17]	Matala-aho, T., A geometric face of Diophantine analysis, (Diophantine Analysis, Trends Math., (2016), Springer), 129-174 · Zbl 1416.11121
[18]	Nesterenko, Yu. V., Hermite-Padé approximants of generalised hypergeometric series, Russian Acad. Sci. Sb. Math., 83, I, (1995)
[19]	Nikišin, E. M., On simultaneous Padé approximants, Math. USSR, Sb., 41, 4, (1982) · Zbl 0478.30007
[20]	van der Poorten, A., Some determinants that should be better known, J. Aust. Math. Soc. A, 21, 278-288, (1976) · Zbl 0343.15002
[21]	Rotman, J., Advanced modern algebra, (2002), Prentice Hall · Zbl 0997.00001
[22]	Shidlovskiĭ, A. B., Transcendental numbers, (1989), Walter de Gruyter · Zbl 0689.10043
[23]	Siegel, C. L., Über einige anwendungen diophantischer approximationen, Abh. Preuss. Akad. Wiss. Phys.-Math. Kl., 1, 1-70, (1929) · JFM 56.0180.05
[24]	Siegel, C. L., Transcendental numbers, Ann. of Math. Stud., vol. 16, (1949), Princeton University Press Princeton, N.J. · Zbl 0039.04402
[25]	Thue, A., Om en generel i store Hele tal uløsbar ligning, Nor. Vidensk. Selsk. Skr., 7, 1-15, (1908)
[26]	Thue, A., Über annäherungswerte algebraischer zahlen, J. Reine Angew. Math., 135, 284-305, (1909) · JFM 40.0265.01