Small solutions to inhomogeneous linear equations over number fields
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- by Robbin O’Leary and Jeffrey D. Vaaler PDF
- Trans. Amer. Math. Soc. 336 (1993), 915-931 Request permission
Abstract:
We consider a system of $M$ independent, inhomogeneous linear equations in $N > M$ variables having coefficients in an algebraic number field $k$ . We give a best possible lower bound on the inhomogeneous height of a solution vector in ${k^N}$ and determine when a solution exists in ${({\mathcal {O}_S})^N}$, where ${\mathcal {O}_S}$ is the ring of $S$-integers in $k$ . If such a system has a solution vector in ${({\mathcal {O}_S})^N}$, we show that it has a solution $\vec \zeta$ in ${({\mathcal {O}_S})^N}$ such that the inhomogeneous height of $\vec \zeta$ is relatively small. We give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system. Our method uses geometry of numbers over adele spaces and local to global arguments.References
- B. J. Birch, A transference theorem of the geometry of numbers, J. London Math. Soc. 31 (1956), 248–251. MR 79616, DOI 10.1112/jlms/s1-31.2.248
- E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823
- I. Borosh, A sharp bound for positive solutions of homogeneous linear Diophantine equations, Proc. Amer. Math. Soc. 60 (1976), 19–21 (1977). MR 422300, DOI 10.1090/S0002-9939-1976-0422300-8
- I. Borosh and L. B. Treybig, Bounds on positive integral solutions of linear Diophantine equations, Proc. Amer. Math. Soc. 55 (1976), no. 2, 299–304. MR 396605, DOI 10.1090/S0002-9939-1976-0396605-3
- I. Borosh and L. B. Treybig, Bounds on positive integral solutions of linear Diophantine equations. II, Canad. Math. Bull. 22 (1979), no. 3, 357–361. MR 555166, DOI 10.4153/CMB-1979-045-2
- I. Borosh, M. Flahive, and B. Treybig, Small solutions of linear Diophantine equations, Discrete Math. 58 (1986), no. 3, 215–220. MR 831816, DOI 10.1016/0012-365X(86)90138-X
- I. Borosh, M. Flahive, D. Rubin, and B. Treybig, A sharp bound for solutions of linear Diophantine equations, Proc. Amer. Math. Soc. 105 (1989), no. 4, 844–846. MR 955458, DOI 10.1090/S0002-9939-1989-0955458-1 J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, New York, 1959. —, Global fields, Algebraic Number Theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press, London 1967. H. Davenport, On the product of three homogeneous linear forms. III, Proc. London Math. Soc. 45 (1939), 98-125. G. Frobenius, Theorie der linearen formen mit ganzen Coefficienten, J. Reine Angew. Math. 86 (1879), 146-208. P. Gordan, Über den grössten gemeinsamen Factor, Math. Ann. 7 (1873), 443-448.
- P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR 893813 I. Heger, Sitzungsber, Akad. Wiss. Wien (Math.) 21 (1856), 550-560.
- Joseph Heinhold, Verallgemeinerung und Verschärfung eines Minkowskischen Satzes, Math. Z. 44 (1939), no. 1, 659–688 (German). MR 1545794, DOI 10.1007/BF01210680 W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, vol. 1, Cambridge Univ. Press, 1968.
- John Hunter, The minimum discriminants of quintic fields, Proc. Glasgow Math. Assoc. 3 (1957), 57–67. MR 91309, DOI 10.1017/S2040618500033463 V. Jarnik, Zwei Bemerkungen zur Geometrie de Zahlen, Vëstnik Kràlovské Ceské Společnosit Nauk, 1941.
- Martin Kneser, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen, Math. Z. 61 (1955), 429–434 (German). MR 68536, DOI 10.1007/BF01181357 J. Mayer, Die absolut-kleinsten Diskriminanten der biquadratischen Zahlkörper, S. B. Akad. Wiss. Wien. IIa 138 (1929), 733-742.
- Peter Scherk, Convex bodies off center, Arch. Math. (Basel) 3 (1952), 303. MR 51882, DOI 10.1007/BF01899231 H. J. S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Roy. Soc. London 151 (1861), 293-326 (=Collected Math. Papers, I, pp. 367-409).
- Thomas Struppeck and Jeffrey D. Vaaler, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 493–528. MR 1084199
- Jeffrey D. Vaaler, Small zeros of quadratic forms over number fields, Trans. Amer. Math. Soc. 302 (1987), no. 1, 281–296. MR 887510, DOI 10.1090/S0002-9947-1987-0887510-6
- André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 0427267, DOI 10.1007/978-3-642-61945-8
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 915-931
- MSC: Primary 11D72; Secondary 11H50
- DOI: https://doi.org/10.1090/S0002-9947-1993-1094559-2
- MathSciNet review: 1094559