On Siegel’s lemma.(English)Zbl 0533.10030

Invent. Math. 73, 11-32 (1983); addendum ibid. 75, 377 (1984).
The following problem is fundamental in effecting auxiliary constructions in transcendental number theory and diophantine approximation: one has $$M<N$$ linear equations $$\sum^{N}_{n=1}a_{mn}x_ n=0$$; $$m=1,2,...,M$$ with rational integer coefficients $$a_{mn}$$ not all zero, to be solved for non-zero N-tuples $$x=(x_ 1,...,x_ N)$$ of rational integers with a good upper bound for the $$| x_ n|$$. For more refined applications the $$a_{mn}$$ may be elements of a number field K, whilst the $$x_ n$$ are restricted to integers of a subfield $$k\subseteq K$$; if $$[K:k]=r$$ we then need $$N>Mr.$$
The present paper solves the above problem once-for-all. The authors find and define a canonical height for the given linear system. They prove the existence of N-Mr linearly independent solutions $$x_{\ell}= (x_{1\ell},x_{2\ell},... ,x_{N\ell})$$ of N-tuples of integers of k so that the product of the heights of these solutions is economically bounded. The results obtained depend on J. D. Vaaler’s cube-slicing inequality [Pac. J. Math. 83, 543-553 (1979; Zbl 0465.52011)] and an adèlic generalization of Minkowski’s theorem on successive minima [see R. B. McFeat, Diss. Math. 88 (1971; Zbl 0229.10014)] proved here independently. These methods are considerably more sophisticated than in Dirichlet’s box principle which was traditionally applied in this context.
This paper will influence the future of transcendental number theory and of diophantine approximation.

MSC:

 11J99 Diophantine approximation, transcendental number theory 11J81 Transcendence (general theory) 11H50 Minima of forms 11R56 Adèle rings and groups

Citations:

Zbl 0465.52011; Zbl 0229.10014
Full Text:

References:

 [1] Bombieri, E.: On the Thue-Siegel theorem. Acta Mathematica148, 255-296 (1982) · Zbl 0505.10015 · doi:10.1007/BF02392731 [2] Bombieri, E., Mueller, J.: On effective measures of irrationality for $$\sqrt[r]{{\frac{a}{b}}}$$ and related numbers. J. reine angew. Math. (to appear) · Zbl 0516.10024 [3] Davenport, H.: Minkowski’s inequality for the minima associated with a convex body. Q. J. Math.10, 119-121 (1939) (=Coll. Works, I, 88-90) · doi:10.1093/qmath/os-10.1.119 [4] Estermann, T.: Note on a theorem of Minkowski. J. London Math. Soc.21, 179-182 (1946) · Zbl 0060.12011 · doi:10.1112/jlms/s1-21.3.179 [5] Fischer, E.: Über den Hadamardschen Determinantensatz. Arch. Math. (Basel)13, 32-40 (1908) [6] Gordan, P.: Über den größten gemeinsamen Factor. Math. Annalen7, 443-448 (1873) [7] Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen. Abh. der Preuß. Akad. der Wissenschaften. Phys.-math. Kl. 1929, Nr. 1 (=Ges. Abh., I. 209-266) [8] Skolem, T.: Diophantische Gleichungen. Ergebnisse der Mathematik, Vol. 5. Berlin: Springer 1938 [9] Smith, H.J.S.: On systems of linear indeterminate equations and congruences. Phil. Trans. Roy. Soc. London151, 293-326 (1861) (=Coll. Math. Papers, I, 367-409) · doi:10.1098/rstl.1861.0016 [10] Thue, A.: Über Annäherungswerte algebraischer Zahlen. J. reine angew. Math.135, 284-305 (1909) · doi:10.1515/crll.1909.135.284 [11] Vaaler, J.D.: A Geometric Inequality with Applications to Linear Forms. Pacific J. Math.83, 543-553 (1979) · Zbl 0465.52011 [12] Vaaler, J.D., van der Poorten, A.J.: Bounds for solutions of systems of linear equations. Bull. Australian Math. Soc.25, 125-132 (1982) · Zbl 0466.10022 · doi:10.1017/S0004972700005104 [13] Weil, A.: Arithmetic on algebraic varieties. Annals of Math.53, 412-444 (1951) (=Coll. Papers, I, 454-486) · Zbl 0043.27002 · doi:10.2307/1969564 [14] Weil, A.: Basic number theory. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0267.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.