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On the inhomogeneous minimum of the product of n linear forms

Published online by Cambridge University Press:  26 February 2010

B. J. Birch  and
H. P. F. Swinnerton-Dyer
B. J. Birch
Affiliation:
Trinity College, Cambridge.
H. P. F. Swinnerton-Dyer
Affiliation:
Trinity College, Cambridge.
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Extract

The following well-known conjecture is generally attributed to Minkowski:

Let L1, …, Ln be n real homogeneous linear forms of determinant Δ ≠ 0 in the n variables x1, …, xn; and let (x1′, …, xn′) be any point. Then there exists a point (x1, xn) congruent to (x1′, …, xn′) (mod 1) at which

Type
Research Article
Information
Mathematika , Volume 3 , Issue 1 , June 1956 , pp. 25 - 39
Copyright
Copyright © University College London 1956

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References

1.Davenport, H., “A simple proof of Remak's theorem on the product of n linear forms”, J. London Math. Soc. 14 (1939), 47–51.CrossRefGoogle Scholar
2.Davenport, H., “On a theorem of Tschebotareff”, J. London Math. Soc., 21 (1946), 28–34.CrossRefGoogle Scholar
3.Dyson, F. J., “On the product of four non–homogeneoua linear forma”, Annals of Math. (2), 49 (1948), 82–109.CrossRefGoogle Scholar
4.Mahler, K., “On lattice points in n–dimensional star bodies. I. Existence theorems”, Proc. Royal Soc. A, 187 (1946), 151–187.Google Scholar
5.Mordell, L. J.Tschebotareff's theorem on the product of non–homogeneous linear forms”, Vierteljahresschrift der Nat. Ges. in Zürich, 85 (1940), Beiblatt, Festschrift R. Fueter, 47–50.Google Scholar
6.Mordell, L. J. “Some Diophantine inequalities”, Mathematika, 2 (1955), 145–149.CrossRefGoogle Scholar
7.Remak, R., “Verallgemeinerung eines Minkowskischen Satzes”, Math. Zeitschr., 17 (1923), 1–34; 18 (1923), 173–200.CrossRefGoogle Scholar
8.Tschebotareff, N. G.; an account is given by Hardy and Wright, Theory of Numbers (2nd ed., Oxford 1945), 396–398.Google Scholar