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Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

Published online by Cambridge University Press:  12 March 2014

H. Luckhardt
H. Luckhardt*
Affiliation:
Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 6000 Frankfurt am Main, West Germany
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Abstract

A previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs of Σ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, where α is (real) algebraic of degree d ≥ 2 and ∣α − pq−1∣ < q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof are polynomial of low degree (in ε−1 and log d). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log log qν, < Cα, εν5/6+ε, where qν are the denominators of the convergents to the continued fraction of α.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 234 - 263
Copyright
Copyright © Association for Symbolic Logic 1989

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References

LITERATUR

[B1] Baker, A., Recent advances in transcendence theory, Proceedings of the Steklov Institute of Mathematics, vol. 132 (1973), pp. 73–76.Google Scholar
[B2] Baker, A., Transcendental number theory, Cambridge University Press, Cambridge, 1975.CrossRefGoogle Scholar
[Bo] Bombieri, E., On the Thue-Siegel-Dyson theorem, Acta Mathematica, vol. 148 (1982), pp. 255–296.CrossRefGoogle Scholar
[Ch] Choodnovsky, G. V., Formules d'Hermite pour les approximants de Padé de logarithmes et de fonctions binômes, et mesures d'irrationalité. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Série A, vol. 288 (1978), pp. 965–967.Google Scholar
[D] Davenport, H., A note on Thue's theorem, Mathematika, vol. 15 (1968), pp. 76–87.CrossRefGoogle Scholar
[D, R] Davenport, H. and Roth, K. F., Rational approximations to algebraic numbers, Mathematika, vol. 2(1955), pp. 160–167.CrossRefGoogle Scholar
[Dy] Dyson, F. J., The approximation to algebraic numbers by rationals, Acta Mathematica, vol. 79 (1947), pp. 225–240.CrossRefGoogle Scholar
[E, V] Esnault, H. and Viehweg, E., Dyson's lemma for polynomials in several variables (and the theorem of Roth), Inventiones Mathematicae, vol. 78 (1984), pp. 445–490.CrossRefGoogle Scholar
[Ev1] Evertse, J.-H., Upper bounds for the numbers of solutions of diophantine equations, Proefschrift, Mathematisch Centrum, Amsterdam, 1983.Google Scholar
[Ev2] Evertse, J.-H., On equations in S-units and the Thue-Mahler equation, Inventiones Mathematicae, vol. 75(1984), pp. 561–584.CrossRefGoogle Scholar
[F] Fel'dman, N. J., An effective refinement of the exponent in Liouville's theorem, Mathematics of the USSR—lzvestija, vol. 5 (1971), pp. 985–1002.CrossRefGoogle Scholar
[G] Gel'fond, A. O., Transcendental and algebraic numbers, Dover, New York, 1960.Google Scholar
[Gi] Girard, J.-Y., Proof theory and logical complexity, Vol. 1, Bibliopolis, Naples, 1987.Google Scholar
[H, B] Hilbert, D. and Bernays, P., Grundlagen der Mathematik. II, 2. Aufl., Springer-Verlag, Berlin, 1970.CrossRefGoogle Scholar
[Hyl] Hyyrö, S., Über das Catalansche Problem, Annales Universitatis Turkuensis, no. 79 (1964).Google Scholar
[Hy2] Hyyrö, S., Über rationale Näherungswerte algebraische Zahlen, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, no. 376 (1965).Google Scholar
[Hy3] Hyyrö, S., Über die Gleichung axn – byn = z und das Catalansche Problem, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, no. 355 (1964).Google Scholar
[K1] Kreisel, G., On the concept of completeness and interpretation of formal systems, Fundamenta Mathematicae, vol. 39 (1952), pp. 103–127.CrossRefGoogle Scholar
[K2] Kreisel, G., Sums of squares, Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd ed., Institute for Defense Analyses, Princeton, New Jersey, 1960, pp. 313–320.Google Scholar
[K3] Kreisel, G., Hilbert's programme and the search for automatic proof procedures, Symposium on automatic demonstration (Versailles, 1968), Lecture Notes in Mathematics, vol. 125, Springer-Verlag, Berlin, 1970, pp. 128–146.CrossRefGoogle Scholar
[K4] Kreisel, G., What have we learnt from Hubert's second problem?Mathematical developments arising from Hilbert problems. Proceedings of Symposia in Pure Mathematics, vol. 28, American Mathematical Society, Providence, Rhode Island, 1976, pp. 93–130.CrossRefGoogle Scholar
[K5] Kreisel, G., On the kind of data needed for a theory of proofs. Logic Colloquium '76, North-Holland, Amsterdam, 1977, pp. 111–128.Google Scholar
[K6] Kreisel, G., Finiteness theorems in arithmetic: an application of Herbrand's theorem for ∑2-formulas, Proceedings of the Herbrand symposium (Marseille, 1981), North-Holland, Amsterdam, 1982, pp. 39–55.CrossRefGoogle Scholar
[K7] Kreisel, G., Mathematical significance of consistency proofs, this Journal, vol. 23 (1958), pp. 155–182.Google Scholar
[K8] Kreisel, G., On the interpretation of non-finitist proofs. Part 1, this Journal, vol. 16 (1951), pp. 241–267.Google Scholar
[K, K] Kreisel, G. and Krivine, J. L., Modelltheorie, Springer-Verlag, Berlin, 1972.CrossRefGoogle Scholar
[K, L] Kreisel, G. and Levy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97–142.CrossRefGoogle Scholar
[K, M] Kreisel, G. and Macintyre, A., Constructive logic versus algebraization. I, Proceedings of the L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), North-Holland, Amsterdam, 1982, pp. 217–260.CrossRefGoogle Scholar
[L, Ma] Lewis, D. J. and Mahler, K., Representations of integers by binary forms, Acta Arithmetica, vol. 6 (1960/61), pp. 333–363.CrossRefGoogle Scholar
[Ma] Mahler, K., An inequlity for the discriminant of a polynomial, Michigan Mathematical Journal, vol. 11 (1964), pp. 257–262.CrossRefGoogle Scholar
[Mi1] Mignotte, M., Quelques remarques sur l'approximation rationnelle des nombres algébriques, Journal für die Reine und Angewandte Mathematik, vol. 268/269 (1974), pp. 341–347.Google Scholar
[Mi2] Mignotte, M., Some useful bounds, Computer algebra. Symbolic and algebraic computation (Buchberger, B.et al., editors), Computing, suppl. 4, Springer-Verlag, Vienna, 1982, pp. 259–263.Google Scholar
[Mue] Mueller, J., On Thue's principle and its applications, Number theory, Noordwijkerhout 1983, Lecture Notes in Mathematics, vol. 1068, Springer-Verlag, Berlin, 1984, pp. 158–166.CrossRefGoogle Scholar
[P1] Perron, O., Algebra. I, II, Teubner, Leipzig, 1927.Google Scholar
[P2] Perron, O., Die Lehre von den Kettenbrüchen. I, Teubner, Leipzig, 1954.Google Scholar
[R] Roth, K. F., Rational approximations to algebraic numbers, Mathematika, vol. 2 (1955), pp. 1–20, 168.CrossRefGoogle Scholar
[Schi] Schinzel, A., Review of [Hy3], Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 137 (1967), p. 258.Google Scholar
[Schm] Schmidt, W. M., Approximation to algebraic numbers, L'Enseignement Mathématique, ser. 2, vol. 17 (1971), pp. 187–1253 = Monographie no. 19 de L'Enseignement Mathématique, Secrétatiat de L'Enseignement Mathématique, Université de Genève, Geneva, 1972.Google Scholar
[Sehn] Schneider, Th., Über die Approximation algebraischer Zahlen, Journal für die Reine und Angewandte Mathematik, vol. 175 (1936), pp. 182–192.CrossRefGoogle Scholar
[S1] Siegel, C. L., Über den Thueschen Satz, Skrifter utgit at Videnskapsselskapet i Kristiania I: Matematisk-Naturvidenskabelig Klasse, 1921, no. 16; reprinted in his Gesammelte Abhandlungen. I, Springer-Verlag, Berlin, 1966, pp. 103–112.Google Scholar
[S2] Siegel, C. L., Über Näherungswerte algebraischer Zahlen, Mathematische Annalen, vol. 84 (1921), pp. 80–99; reprinted in his Gesammelte Abhandlungen. I, Springer-Verlag, Berlin, 1966, pp. 77–96.CrossRefGoogle Scholar
[St] Statman, R., Lower bounds on Herbrand's theorem, Proceedings of the American Mathematical Society, vol. 75 (1979), pp. 104–107.Google Scholar
[Th] Thue, A., Über Annäherungswerte algebraischer Zahlen, Journal für die Reine und Angewandte Mathematik, vol. 135 (1909), pp. 284–305.CrossRefGoogle Scholar
[T] Tijdeman, R., On the equation of Catalan, Acta Arithmetica, vol. 29 (1976), pp. 197–209.CrossRefGoogle Scholar