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On the product of three non-homogeneous linear forms

Published online by Cambridge University Press:  24 October 2008

H. Davenport
H. Davenport
Affiliation:
University CollegeLondon
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Extract

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbers a, b, c there exist integral values of u, v, w for which

and the constant ⅛ on the right is best possible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 2 , April 1947 , pp. 137 - 152
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

* J. London Math. Soc. 14 (1939), 47–51.Google Scholar

† When making this suggestion, Prof. Mordell told me that he had proved (3) for the linear forms (4) with a value of appreciably less than ⅛.

* See Davenport, , Proc. Cambridge Phil. Soc. 39 (1942), 1–21CrossRefGoogle Scholar and references given there.

† Proc. K. Akad. Wet. Amsterdam, 49 (1946), 815–821.Google Scholar

* The field k(θ) is, of course, galoisian, and is transformed into itself by θ → ø, which implies ø → ψ and ψ → θ.

† We name θ ø, ψ so that

* See the previous footnote.

* This field has the same properties as those stated for k(θ) in an earlier footnote.

† We name θ′ ø′, ψ′ so that